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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 30, Iss. 12 — Dec. 1, 2013
  • pp: 2519–2525
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Generalization of the optical theorem for light scattering from a particle at a planar interface

Alex Small, Jerome Fung, and Vinothan N. Manoharan  »View Author Affiliations


JOSA A, Vol. 30, Issue 12, pp. 2519-2525 (2013)
http://dx.doi.org/10.1364/JOSAA.30.002519


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Abstract

The optical theorem provides a powerful tool for calculating the extinction cross section of a particle from a solution to Maxwell’s equations, relating the cross section to the scattering amplitude in the forward direction. The theorem has been generalized by a number of other workers to consider a particle near an interface between media with different refractive indices. Here we present a derivation of the generalized optical theorem that is valid for a particle embedded in the interface, as well as an incident beam undergoing total internal reflection. We also obtain an additional useful physical result: we show that the far-field scattered field must be zero in the direction parallel to the interface. Our results enable the verification of computations of scattering by particles embedded in interfaces and may be relevant to experiments on colloidal particles at fluid interfaces.

© 2013 Optical Society of America

1. INTRODUCTION

The optical theorem [1

1. M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

3

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

] is a powerful result in scattering theory, relating the extinction cross section σe of a particle to the scattering amplitude in the forward direction. This theorem is particularly useful for checking or applying the results of light scattering codes [4

4. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

6

6. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]

], since for a nonabsorbing particle, the integral of the scattered power over 4π of solid angle must be proportional to the imaginary part of the forward scattering amplitude. For an absorbing particle, the theorem can be used to obtain σe from forward scattering, and analytical integration or numerical quadrature can be used to obtain the scattering cross section σs from the scattered field in all directions. The difference between σe and σs is then the absorption cross section σa. For any particle, extinction is a modification (not necessarily a reduction [7

7. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008). [CrossRef]

,8

8. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009). [CrossRef]

]) of the amplitude of the wave propagating in the forward direction arising from both scattering and absorption. Extinction can be viewed as a manifestation of interference between the incident wave and the light scattered in the forward direction.

The optical theorem is usually presented for a particle in a homogeneous medium. Situations such as backscattering from dust on optical mirrors and coatings have also motivated theoretical and experimental studies of scattering by objects on or near planar interfaces [9

9. K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987). [CrossRef]

13

13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005). [CrossRef]

]. The problem of scattering by an object embedded at an interface has received far less attention, but is relevant to systems such as colloidal particles interacting with liquid–liquid interfaces [14

14. D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012). [CrossRef]

]. In experiments on these systems, the objective is to study the position and motion of the particle itself, and scattering solutions are required to extract the particle position from a far-field interferometric measurement. But because of the difficulty of modeling scattering from a particle at a refractive index discontinuity, the experiments typically use index-matched liquids. A scattering solution for a particle at an interface would permit a broader range of systems to be examined and enable new experimental measurements, such as the in situ determination of the contact angles of submicrometer particles. The first step toward such a solution is the generalization of the optical theorem for a particle embedded at an interface.

Fig. 1. (a) Schematic and (b) coordinate system for the particle and interface, as well as the directions of the incident, reflected, transmitted, and scattered fields.

Building on Torrungrueng’s work, we consider scattering from an embedded particle, considering illumination both with and without total internal reflection. We obtain an optical theorem identical to that of Torrungrueng when there is no total internal reflection. Furthermore, our approach leads to an additional physical insight: it reveals a boundary condition that must be satisfied by the scattered field for the scattered power to be well-defined in the far field. While this condition could also be derived by attempting to match fields at the interface, and is indeed observed to be satisfied in numerical solutions to scattering problems at interfaces [17

17. A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

], we show that it is necessary for the optical theorem to be valid. Our generalized optical theorem and boundary condition may be used to verify numerical computations and experimental measurements of scattering by particles embedded at interfaces under a wide range of conditions.

We begin by assuming that we have an isolated particle embedded at a planar interface between two dielectric, nonabsorbing media. The particle may have arbitrary size, shape, and composition, and may be absorbing. We assume that the particle is illuminated by a plane wave of infinite extent propagating at an arbitrary angle relative to the interface. We then, like Torrungrueng et al. [16

16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004). [CrossRef]

], compute the total electromagnetic power passing through a large imaginary sphere surrounding the particle, which we relate to the extinction cross section. Careful consideration of the integrals needed to compute the power, for both ordinary and totally internally reflected illumination, then leads to generalizations of the optical theorem and a new boundary condition on the far-field scattered field at the interface. Finally, we discuss the physical nature of our new boundary condition using several additional arguments.

2. DEFINITIONS AND FORMALISM

A. Geometry

In our coordinate system (Fig. 1(b)), the plane z=0 corresponds to the interface between two media of indices n1 (z<0) and n2 (z>0), and corresponding magnetic permeabilities μ1 and μ2. We assume that a scattering particle (which may also absorb electromagnetic radiation) straddles the interface. While Fig. 1 is drawn such that the particle has its approximate center at the origin, we do not assume this.

B. Incident, Reflected, and Transmitted Fields

We assume the system to be illuminated by a plane wave coming from the first medium (n1,μ1), propagating in the +z direction. Its wave vector in the first medium is ki=n1k0(cosθiz^+sinθix^); that is, it travels in the (x,z) plane. We assume that the incident electric field has vector amplitude E^1,i. The magnitude of E^1,i is unity (for convenience), and the direction indicates the polarization. We first consider the case of no total internal reflection. Thus, when the wave hits the interface, it is partially reflected and partially transmitted. The amplitudes of the reflected (E^1,r) and transmitted (E^2,t) electric fields are determined by the Fresnel coefficients for the TE and TM components, and the directions by the Law of Reflection and Snell’s Law. The associated magnetic fields are determined from Faraday’s Law:
×E=1cμHtik×E=ik0μH,H=kμk0×E.
(1)

The incident power is determined from the magnitude of the time-averaged Poynting vector S=(c/2)Re(E×H*). For notational convenience, we leave out the time-averaging brackets in what follows, and we let c=1. Because |B|=n|E|=μ|H|, and the incident field has unit amplitude, the incident power per unit area is
Iinc=n12μ1.
(2)
This relation is needed at the end of the derivation, when we divide the scattered power by incident intensity to determine the cross section.

To be consistent with derivations of the optical theorem in homogeneous media, we refer to the incident, reflected, and transmitted fields collectively as Ei and Hi. While the reflected and transmitted fields are not, of course, “incident,” they are independent of the particle. For clarity, we will henceforth refer to the incident field in the traditional sense as the incoming field.

C. Scattered Fields

In the far field (r much larger than λ and the size of the particle), the scattered fields Es and Hs take the asymptotic form
Es=ξ(θ,ϕ)exp(ink0r)k0r=(ξθ(θ,ϕ)θ^+ξϕ(θ,ϕ)ϕ^)exp(ink0r)k0r,
(3)
Hs=nμr^×Es,
(4)
where ξ(θ,ϕ) is the vector scattering amplitude for scattering in the (θ,ϕ) direction, and we have applied Faraday’s Law to determine Hs. We suppress the implied eiωt dependence of all fields for convenience. Because the radiated fields are transverse, ξ has only θ and ϕ components. We assume that ξ exactly describes the far-field scattering of the given particle at the interface of the two media. ξ thus explicitly includes the effects of the interface via its angular dependence.

3. CALCULATING THE SCATTERED POWER

A. Form of the Key Terms

We begin by considering the energy passing through a large imaginary sphere of radius r, centered at the origin and surrounding the particle. We assume k0r1, so that the scattered fields on the sphere are in the far field. The rate at which energy is absorbed inside this sphere is given by the negative of the flux of the Poynting vector through the sphere:
Wa=12Re(4π(E×H*)·r^r2dΩ)=12Re(4π(Ei×Hi*+Es×Hs*+Ei×Hs*+Es×Hi*)·r^r2dΩ),
(5)
where we have left out time-averaging brackets, since the effects of averaging are encompassed in the 1/2 factor. The first term is the flux due to the incoming, reflected (from the interface) and transmitted (through the interface) beams. Since medium 1 and medium 2 are assumed to be nonabsorbing, this term integrates to zero. The second term is the negative of the scattered power Ws. The two remaining terms are due to interference between the scattered field and the other fields, which manifests as extinction. Thus,
Wa+Ws=Wext=12Re(4π(Ei×Hs*+Es×Hi*)·r^r2dΩ).
(6)

Since |k|/k0=n, the first term on the right side of Eq. (6) simplifies to
r^·(Ei×Hs*)=Ei·(r^×Hs*)=Ei·(r^×(r^×nμEs*))=nμEi·Es*.
(7)
The first step follows from the triple scalar product rule. In the last step, the two cross products of the scattered field with r^ yield the scattered field (with a minus sign) because r^ is perpendicular to the field and has unit magnitude.

The second term of Eq. (6) is similar to the first term, but with the electric and magnetic fields interchanged. We now show that this term has the same magnitude as the first term. In the far field, the scattered electric and magnetic fields, like the incident fields, are both transverse to the direction of propagation, and are mutually perpendicular to each other. Their magnitudes are also equal, up to a factor of n. Consequently, swapping Ei for Hi and Es for Hs does not change the magnitude of the cross product, as one swap gives a factor of n and the other gives a factor of 1/n, resulting in cancellation under multiplication. (This is only true in the far field; in the near field, the electric and magnetic fields can decouple.) Alternately, if one thinks of the fields as superpositions of plane waves in Fourier space, swapping the electric and magnetic fields amounts to flipping all polarizations, which does not change any relative orientations. Consequently, in the far field, both vector terms in the integrand of Eq. (6) should have equal magnitude.

However, when electric and magnetic fields are interchanged, there is a sign ambiguity that needs to be considered for determining the sign of the dot product with r^. In anticipation of the key result of this paper—that the values of the key integrals, Eq. (6), are proportional to the value of the scattering amplitude in the directions of the “incident” plane waves probing the particle—we consider the scattering amplitude in the second term of Eq. (6) for those directions.

There are two physical possibilities that result in different signs: the “incident” plane wave and the scattered wave can propagate in either the same direction or in opposite directions. For Ei and Hi arising from the transmitted or reflected plane waves, in the far field, the scattered field and the relevant plane wave both propagate in the same direction. Thus, Ei×Hs* and Es×Hi* both yield vectors pointing in the same direction, and as already argued, both of these cross products have the same magnitude (up to complex conjugation) [Fig. 2(a)]. At most, scattering can rotate the scattered field relative to the “incident” field, so that the angle between Es and Hi is the same as the angle between Ei and Hs. In the direction of the incoming plane wave, however, the scattered field and incoming field transport energy in opposite directions. Consequently, the angle between Es and Hi differs from the angle between Ei and Hs by 180°, and swapping the fields introduces a minus sign in the cross product [Fig. 2(b)].

Fig. 2. Mutual orientations of Ei and Hs, and Es and Hi, for “incident” and scattered fields that are copropagating or counterpropagating. All vectors in each part exist at the same physical point, but have been separated for clarity. (a) The “incident” and scattered fields propagate in the same direction, so both Ei×Hs and Es×Hi point in the same direction. (b) The “incident” and scattered fields propagate in opposite directions; Ei×Hs and Es×Hi point in opposite directions.

We therefore have the following relation for the second term in Eq. (6):
r^·Es×Hi*=±nμEi*·Es,
(8)
that is, plus (for the reflected and transmitted waves) or minus (for the incoming wave) the complex conjugate of the result in Eq. (7) for the first term. When both terms in Eq. (6) have the same sign, adding Eq. (7) to its complex conjugate will give twice its real part. When the second term has a minus sign, we will get 2i times the imaginary part of Eq. (7). Ultimately, after evaluating the integrals in Eq. (6), we will take the real part to get a physical extinction power, and the imaginary part will not contribute. Consequently, in the integrals in Eq. (6), rather than having to consider three contributions to Ei, we need only consider the contributions from the transmitted and reflected plane waves. We need not consider contributions to Ei from the incoming wave.

B. Key Integral

To evaluate the integral in Eq. (6), we consider each medium separately. We consider only the first term, since the second term has equal magnitude, as argued above. For convenience, we change the coordinate system from that shown in Fig. 1. We will separately consider the contributions to Ei from the transmitted and reflected plane waves, and for each of these plane waves choose the z axis to be parallel to the direction of propagation. This makes it easy to express the plane waves as eink0rcosθ, but complicates the limits of integration. The outgoing fields remain in the form (ξ(θ,ϕ)/k0r)expink0r.

Referring to Eq. (6), we thus have to compute integrals of the form
mediumEi·Es*r2dΩ=ϕinterfacecosθ=11k0rξ*(θ,ϕ)·E0eink0r(1cosθ)r2dcosθdϕ,
(9)
where E0 is the amplitude of the appropriate plane wave. The lower limit on θ depends on the value of ϕ, as shown in Fig. 3. The second step follows from our physical assumption that the most slowly decaying terms in the scattered field decay no more gradually than 1/r. There will, in general, be near-field terms that decay more rapidly than 1/r, but those terms can be ignored at large r. Were the scattered fields to decay more slowly than 1/r, the scattered power would increase without bound as the radius of the sphere over which we compute the power increases.

Fig. 3. Polar angle θ (measured with respect to the k vector) at the interface depends on the azimuthal angle ϕ.

We first do the integral over cosθ, integrating by parts:
mediumEi·Es*r2dΩ=ϕξ*(θ,ϕ)·E0eink0r(1cosθ)ink02|interfacecosθ=1dϕϕinterfacecosθ=11ink02ξ*(θ,ϕ)cosθ·E0eink0r(1cosθ)dcosθdϕ.
(10)
Only the first term in Eq. (10), the boundary term from integrating by parts, survives in the far field. In the boundary term, a factor of 1/r from the asymptotic dependence of the fields and an additional factor of 1/r from integrating the exponential together cancel the factor of r2 from integrating over area. The second term of Eq. (10) vanishes in the far field: further integration by parts would result in two more terms each with an additional factor of 1/r.

The boundary term in Eq. (10) is straightforward to evaluate at the upper bound. Subsequently, integration over ϕ—around the pole in spherical coordinates—gives a factor of 2π, since ξ is independent of ϕ at θ=0. We therefore obtain
mediumEi·Es*r2dΩ=2πξ*(θ,ϕ)·E0ink02+interface term.
(11)
The first term on the right comes from the upper bound, and there is an additional contribution from evaluating the boundary term in Eq. (10) at the lower bound, which occurs at the interface.

Evaluating the interface term results in a serious problem. Components of the “incident” and scattered fields parallel to the interface must be continuous across the interface. The Fresnel coefficients enforce this condition for the “incident” plane waves, and we require ξ to be continuous across the interface. However, expink0r(1cosθ) is not continuous across the interface. Consequently, the terms in each medium arising from evaluating Eq. (10) at the interface do not cancel each other unless the media have the same refractive index. We thus have a result for the extinction power that depends on r, which is clearly unphysical.

It is tempting to assume that these terms at the interface vanish upon integration over ϕ, since the integrand contains a rapidly oscillating exponential. But for incident beams that are normal or nearly normal to the interface, θ is approximately 90° at the interface, and the 1cosθ factor in the exponent (approximately 1) depends only weakly on ϕ when integrating around the interface. Integrating over ϕ only gives a vanishing result if expink0r(1cosθ) oscillates more rapidly than ξ(θ,ϕ) as a function of ϕ. For an off-normal incoming wave and r this condition is satisfied, but not for a normal incoming wave. Likewise, for an incoming wave that is close to normal incidence, the extinction power could oscillate as a function of r at distances out to λ/Δθ (where Δθ is the deviation from normal incidence). For small Δθ this oscillation of the extinction power could thus persist at distances typically associated with the far field. Consequently, we have the potential for unphysical oscillatory dependence of the extinction power on r in the far field.

The only resolution of this problem is to require that ξ be zero at the interface. Additional arguments to help understand this physical condition are given in Section 5. This condition does not mean that the scattered fields vanish at the interface; nonradiative components could still exist in the near field. Using this condition, we conclude that the value of the key integral, Eq. (9), is given (in the limit of large r) by Eq. (11):
mediumEi·Es*r2dΩ=2πink02ξ*(θ,ϕ)·E0.
(12)

C. Generalized Optical Theorem

To go from the key integral, Eq. (12), to the generalized optical theorem, we apply the following steps. We already showed in Subsection 3.A that we only need to consider the reflected and transmitted contributions to the “incident” wave Ei. From Eq. (6) and the results of Subsection 3.A, for these two contributions, we add the complex conjugate of the result in Eq. (12), yielding twice the real part. This factor of 2 cancels a factor of 1/2 from computing a time average. Moreover, we have a prefactor of n/μ in front of the integrand in the key integral. We thus obtain
Wext=2πk02Re(1iμ2ξ*(kt)·E2,t+1iμ1ξ*(kr)·E1,r)=2πk02Im(1μ2ξ*(kt)·E2,t+1μ1ξ*(kr)·E1,r).
(13)
Here, E1,r is the reflected plane wave in medium 1 and E2,t is the transmitted plane wave in medium 2. Since the Fresnel coefficients are real, the “incident” fields have real amplitudes, and we can re-express this in terms of ξ at the expense of a minus sign:
Wext=2πk02Im(1μ2ξ(kt)·E2,t+1μ1ξ(kr)·E1,r).
(14)

To go from power to cross section we divide by the intensity of the incident beam, which is μ1/2n1, according to Eq. (2). We find that the extinction cross section is
σe=4πn1μ1k02Im(1μ2ξ(kt)·E2,t+1μ1ξ(kr)·E1,r).
(15)
This result, our generalized optical theorem, is equivalent to that of Torrungrueng et al. [16

16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004). [CrossRef]

] and reduces to the traditional optical theorem in the absence of an interface, when there is no reflected plane wave. The incoming polarization and the Fresnel coefficients determine E1,r and E2,t. Other conventions for the assumed forms of the “incident” plane waves and scattered wave could lead to slightly different factors of n, 2, and π, and possibly the replacement of the real part with the imaginary part, depending on the definition of ξ. The key point remains that the extinction cross section is determined by the scattering amplitude in the directions of the transmitted and reflected plane waves.

4. TOTAL INTERNAL REFLECTION

We now consider the possibility that the particle is illuminated by a plane wave that is totally internally reflected from the interface. In this case, the transmitted beam is evanescent in the second medium and has the form eikxxκz, where kx=n1k0sinθi and κ=k0n12sin2θin22. Here we are reverting to a coordinate system with the z axis perpendicular to the interface between medium 1 and medium 2.

Our generalized optical theorem for ordinary illumination, Eq. (15), shows that σe depends on the phase of the scattered field relative to the “incident” fields in the directions in which electromagnetic energy is transported in the absence of a particle. In the traditional optical theorem, we consider the forward direction, and for our generalized optical theorem, we consider the directions of the transmitted and reflected plane waves. When there is total internal reflection, only the reflected wave transports energy in the absence of a particle. It would seem natural that the relevant generalization of the optical theorem for total internal reflection should depend only on the scattering amplitude in the direction of the reflected plane wave. We now show that this is indeed the case by examining in detail the key integral, Eq. (9), in medium 2. This is necessary because the form of the evanescent wave precludes the straightforward application of our work in Subsection 3.B.

In medium 2, Eq. (9) becomes
medium 2Ei·Es*r2dΩ=ϕ=02πcosθ=01eκrcosθeikxrsinθcosϕξj*(θ,ϕ)ein2k0rk0rr2d(cosθ)dϕ=z=0reκzϕ=02πξj*(θ,ϕ)ein2k0rk0eikxr2z2cosϕdϕdz,
(16)
where ξj is some component of ξ (the specific component depending on the incident polarization), and we have changed variables from cosθ to z=rcosθ.

We now argue that for fixed z, the inner integral on the right vanishes. The exponential oscillates rapidly as a function of ϕ for large r (unless zr, in which case the eκz factor goes to zero as r becomes large). However, when we consider ξj as a function of ϕ and compute its Fourier series over the range 0ϕ2π, at large spatial frequencies the amplitudes must go to zero, an assumption we justify below. Consequently, the exponential oscillates far more rapidly than ξj, and the integral of a slowly varying function with a rapidly oscillating sinusoidal function with no constant offset is zero.

One can see that the high-frequency amplitudes in the Fourier series for ξj(ϕ) go to zero from two different arguments. The first argument comes from considering physical models of scattering in limiting cases: For small particles and/or small refractive index contrasts, the angular width of the scattering profile scales as λ/a, where a is a characteristic dimension. For aλ, the angular width is large, and consequently the scattering peak in Fourier space is narrow, scaling as a/λ. Thus, high-frequency components in the Fourier spectrum are negligible.

A large particle, of course, can be treated with geometrical optics for many purposes, and in geometrical optics one can obtain specular reflections that produce sharp peaks in the angular spectrum. However, the finite size a of the reflecting surface means that the peaks are ultimately diffraction-limited, with angular widths scaling as λ/a and widths in frequency space scaling as a/λ. At larger frequencies, the Fourier spectrum of the far-field scattering amplitude ξ goes to zero, assuming that we consider measurements at a distance ra.

The second argument is that the scattered power is finite. The integral of ξj2 over ϕ is, by Parseval’s theorem, equal to the sum of the squares of the coefficients in the Fourier series for ξj. The integral of ξj2 must be finite, so the sum of squared coefficients must converge. Consequently, the coefficients must decay to zero for sufficiently high frequencies.

Given that the inner integral is zero, in the presence of total internal reflection there is no transmission contribution to the extinction cross section, and our generalized optical theorem is
σe=4πn1μ12k02Im(ξ*(kr)·E1,r).
(17)
We have thus justified our intuition that the strength with which the particle modifies the specularly reflected field determines the extinction cross section.

The evanescent field probing the portion of the particle in the second medium nonetheless contributes to the extinction cross section. In particular, the local field in the second medium contributes to the far-field scattering in the first medium. This is most clearly apparent when a Green’s function is used to compute the far-field scattering given the field inside the particle, following Carney et al. [12

12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004). [CrossRef]

]: the Green’s function integral extends over the entire particle, including the portion embedded in the second medium. Thus, the evanescent field still contributes to ξ in the direction of the reflected wave.

Eq. (17) differs from our result for ordinary illumination beyond the absence of the term corresponding to the transmitted wave: Eq. (17) includes a complex conjugate that is not found in Eq. (15). The difference in the complex conjugate is physically insignificant. It arises because totally internally reflected plane waves are phase-shifted, and the Fresnel coefficients are complex. In Eq. (15), because the Fresnel coefficients for ordinary reflection are real, we could equate Im(ξ*·Ei) with Im(ξ·Ei). Nonetheless, our work in this section demonstrates that in both Eqs. (15) and (17), the extinction cross section is proportional to the sine of the phase shift between the “incident” and scattered fields.

5. TWO OTHER ARGUMENTS FOR ZERO FAR-FIELD SCATTERING ALONG THE INTERFACE

We argued above that the far-field scattering amplitude ξ is zero along the interface. This came from the requirement that the integrated scattered power be independent of r for rλ and near-normal incident fields. The key integrals include interfacial terms that do not cancel because they depend on expink0r, which is not continuous across the interface. Another way to understand this is from the continuity of the tangential components of the fields. The asymptotic form of the scattered field must be (ξ(θ,ϕ)/k0r)expink0r, yet the exponential is inherently discontinuous across the interface. The only solution is for ξ(θ,ϕ) to be zero at the interface. We emphasize that the scattered field is only zero far from the particle, where the asymptotic form oscillates as expink0r. In the near field, the scattered field can (and, generally, will) be nonzero.

Having established that ξ must vanish at the interface, we now give a heuristic argument suggesting how this happens. Our argument follows the approach of Carney and co-workers [12

12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004). [CrossRef]

,13

13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005). [CrossRef]

], who treat the scattered field as a superposition of fields from the collection of dipoles excited in the particle. Each dipole radiates anisotropically but in all directions. Consider first a detector near the interface that detects the fields radiated by a dipole in the same medium. The detector will receive both a direct field and a field reflected from the interface. As the detector gets closer and closer to the interface, the angle of incidence for the reflected field gets closer to 90° (grazing incidence). At grazing incidence, the Fresnel reflection coefficient goes to 1, so the direct and reflected fields cancel.

Note that the Fresnel coefficient for reflection at near-grazing angles goes to 1 irrespective of the index mismatch between the media. Consequently, taking the limit |n1n2|1 will not recover the case of a particle in a homogeneous medium. Even with a vanishingly small index mismatch, it is impossible to match fields in the far field, and so the Fresnel reflection coefficient must go to 1 for grazing incidence. However, the angular width of this zero in the profile of scattered light can become smaller as |n1n2|0. Our requirement of zero scattering in the plane of the interface does not impose any lower limit on the angular width of this drop in scattering intensity.

If the dipole and detector are on opposite sides of the interface, the situation is a bit more complicated. If the detector is in a low-index medium, light that hits the interface at the critical angle in the high-index medium can be refracted into the direction parallel to the interface. However, at the critical angle the reflection coefficient has unit magnitude, so there is no flux of energy into the low-index medium. Consequently, no power can be incident on a detector located slightly above the interface in the low-index medium.

Alternately, one could couple an evanescent mode in the low-index medium into the high-index medium, which can sustain the higher spatial frequencies, through a mode that propagates parallel to the interface in the high-index medium. A direct calculation can be done for this case, but the argument is rather tedious [18

18. W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]

]. A simpler argument, one that embraces this case as well as the three other cases (transmission into the low-index medium, reflection in the other medium), involves considering the wave equation for a mode parallel to the interface.

Let us assume a field component with spatial dependence E(z)ei(kxx+kyy) in the high-index medium. The Helmholtz equation can be written as
2+n2(z)k02E=2z2E(kx2+ky2)E+k02n2(z)E=2z2E+k02(n2(z)kx2+ky2k02)=0.
(18)
We can treat n2(kx2+ky2)/k02 as an effective dielectric constant for this mode. If the mode is propagating parallel to the interface then kx2+ky2=nk0, so the effective index is zero. In the low-index medium the effective dielectric constant is nonzero (but negative). Consequently, the effective index contrast between the media is infinite, and the interface causes perfect reflection. At the surface of a perfect reflector the field must be zero, and the assumption of propagation parallel to the interface means that the field is constant, since 2E/z2=0 in the high-index medium. Thus, a mode propagating parallel to the interface cannot be sustained.

We note that this analysis is only meant to provide a qualitative understanding of the mechanism by which the scattered fields at the interface vanish in the far field. A dipole that is inside the particle but very close to the interface could send light to the detector via “reflection” from a point inside the particle itself, which would require a more careful analysis, such as that of Carney et al., who derive a similar version of the optical theorem by separately treating waves reflected and transmitted by the interface. The most rigorous argument for zero far-field scattering in the plane of the interface comes from the impossibility of matching the asymptotic form for the scattered field at the interface.

6. CONCLUSION

In summary, we have derived a generalized optical theorem for electromagnetic waves scattering off a particle at an interface, with an approach that enables a treatment of scattering from a particle that straddles an interface, e.g., a colloidal particle at a liquid–liquid interface. Additionally, we have shown a connection between the optical theorem and an important boundary condition that the scattered field must satisfy far from the particle. This boundary condition, that the scattering amplitude is zero in the direction parallel to the interface, may be useful in checking the results of scattering codes, and supplements the check already provided by the optical theorem. Also, our analysis shows that this boundary condition is not only a consequence of the reflection properties of the interface, it is also a necessary condition for the scattered power to be well-defined in the far field and for the optical theorem to be valid.

ACKNOWLEDGMENTS

A portion of the work was conducted at the Kavli Institute for Theoretical Physics at the University of California, Santa Barbara. This research was supported in part by the National Science Foundation under grants PHY11-25915 and CBET-0747625.

REFERENCES

1.

M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).

2.

H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).

3.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

4.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]

5.

V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002). [CrossRef]

6.

D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]

7.

M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008). [CrossRef]

8.

M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009). [CrossRef]

9.

K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987). [CrossRef]

10.

L. Sung, G. W. Mulholland, and T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999). [CrossRef]

11.

J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. 41, 5405–5412 (2002). [CrossRef]

12.

P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004). [CrossRef]

13.

D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005). [CrossRef]

14.

D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012). [CrossRef]

15.

P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).

16.

D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004). [CrossRef]

17.

A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.

18.

W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]

OCIS Codes
(290.0290) Scattering : Scattering
(290.2200) Scattering : Extinction
(290.2558) Scattering : Forward scattering
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: August 14, 2013
Revised Manuscript: October 17, 2013
Manuscript Accepted: October 17, 2013
Published: November 12, 2013

Virtual Issues
Vol. 9, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Alex Small, Jerome Fung, and Vinothan N. Manoharan, "Generalization of the optical theorem for light scattering from a particle at a planar interface," J. Opt. Soc. Am. A 30, 2519-2525 (2013)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-30-12-2519


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References

  1. M. Born, E. Wolf, and A. B. Bhatia, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University, 1999).
  2. H. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).
  3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  4. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988). [CrossRef]
  5. V. Villamizar and O. Rojas, “Time-dependent numerical method with boundary-conforming curvilinear coordinates applied to wave interactions with prototypical antennas,” J. Comput. Phys. 177, 1–36 (2002). [CrossRef]
  6. D. W. Mackowski, “Calculation of total cross sections of multiple-sphere clusters,” J. Opt. Soc. Am. A 11, 2851–2861 (1994). [CrossRef]
  7. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25, 1504–1513 (2008). [CrossRef]
  8. M. I. Mishchenko, M. J. Berg, C. M. Sorensen, and C. V. van der Mee, “On definition and measurement of extinction cross section,” J. Quant. Spectrosc. Radiat. Transfer 110, 323–327 (2009). [CrossRef]
  9. K. B. Nahm and W. L. Wolfe, “Light-scattering models for spheres on a conducting plane: comparison with experiment,” Appl. Opt. 26, 2995–2999 (1987). [CrossRef]
  10. L. Sung, G. W. Mulholland, and T. A. Germer, “Polarized light-scattering measurements of dielectric spheres upon a silicon surface,” Opt. Lett. 24, 866–868 (1999). [CrossRef]
  11. J. H. Kim, S. H. Ehrman, G. W. Mulholland, and T. A. Germer, “Polarized light scattering by dielectric and metallic spheres on silicon wafers,” Appl. Opt. 41, 5405–5412 (2002). [CrossRef]
  12. P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for scalar fields,” Phys. Rev. E 70, 36611 (2004). [CrossRef]
  13. D. R. Lytle, P. S. Carney, J. C. Schotland, and E. Wolf, “Generalized optical theorem for reflection, transmission, and extinction of power for electromagnetic fields,” Phys. Rev. E 71, 56610 (2005). [CrossRef]
  14. D. M. Kaz, R. McGorty, M. Mani, M. P. Brenner, and V. N. Manoharan, “Physical ageing of the contact line on colloidal particles at liquid interfaces,” Nat. Mater. 11, 138–142 (2012). [CrossRef]
  15. P. S. Carney, “The optical cross-section theorem with incident fields containing evanescent components,” J. Mod. Opt. 46, 891–899 (1999).
  16. D. Torrungrueng, B. Ungan, and J. Johnson, “Optical theorem for electromagnetic scattering by a three-dimensional scatterer in the presence of a lossless half space,” IEEE Geosci. Remote Sens. Lett. 1, 131–135 (2004). [CrossRef]
  17. A. Doicu, R. Schuh, and T. Wriedt, “Scattering by particles on or near a plane surface,” in Light Scattering Reviews 3 (Springer, 2008), pp. 109–130.
  18. W. Lukosz and R. Kunz, “Light emission by magnetic and electric dipoles close to a plane dielectric interface. II. Radiation patterns of perpendicular oriented dipoles,” J. Opt. Soc. Am. 67, 1615–1619 (1977). [CrossRef]

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