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Optics Express

Optics Express

  • Editor: C. Martijin de Sterke
  • Vol. 15, Iss. 9 — Apr. 30, 2007
  • pp: 5572–5588
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Rayleigh-Gans scattering approximation: surprisingly useful for understanding backscattering from disk-like particles

Howard R. Gordon  »View Author Affiliations


Optics Express, Vol. 15, Issue 9, pp. 5572-5588 (2007)
http://dx.doi.org/10.1364/OE.15.005572


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Abstract

Recent computations of the backscattering cross section of randomly-oriented disk-like particles (refractive index, 1.20) with small-scale internal structure, using the discrete-dipole approximation (DDA), have been repeated using the Rayleigh-Gans approximation (RGA). As long as the thickness of the disks is approximately 20% of the wavelength (or less), the RGA agrees reasonably well quantitatively with the DDA. The comparisons show that the RGA is sufficiently accurate to be useful as a quantitative tool for exploring the backscattering features of disk-like particles with complex structure. It is used here to develop a zeroth-order correction for the neglect of birefringence on modeling the backscattering of detached coccoliths from E. huxleyi.

© 2007 Optical Society of America

1. Introduction

For many of the cases examined, I have also computed the backscattering cross section using the Rayleigh-Gans approximation (RGA) to scattering. The RGA is applicable when the relative refractive index of the particle (m) is close to unity, and the “size” is ≪ the wavelength of light divided by |m - 1| [14

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

, 15

15. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

]. Thus the size need not be ≪ the wavelength. It is computationally fast when compared to any other method because analytical formulas are available for many particle shapes. Moreover, extension to particles of any shape is straightforward.

In this paper I compare backscattering by disk-like particles (with refractive index 1.20 relative to water) computed using the RGA with exact (DDA) computations. The comparisons show that the RGA is sufficiently accurate to be useful as a quantitative tool for exploring the backscattering features of disk-like particles with complex structure, e.g., disks with angular periodicities or detached coccoliths from the coccolithophored E. huxleyi. The validity of the RGA for such particles allows investigation of the influence of their birefringence on backscattering.

2. The Electromagnetic scattering problem

Conceptually, the electromagnetic scattering problem can be developed in a simple manner. If a particle is subjected to an incident electromagnetic field E⃗(0)(D⃗i,t), then a volume element dVi at a position D⃗t within the particle (Fig. 1) will experience an electric field E⃗(D⃗i,t) given by

EDit=E(0)Dit+jCDiDjEDjt,
(1)

where the sum excludes i=j. The E’s on both sides of this equation are unknown, while E (0) and C are known functions of position and time. This electric field induces a dipole moment (dp) in dVi given by

dpDit=ρnαEDitdVi,
(2)

where α is the polarizability tensor and ρn is the number density of atoms (molecules). At a great distance r⃗i from the particle the field due to the dipole moment induced in dVi is

dE(s)=14πε0riκ×[κ×dpi(Di,tric)],
(3)

where κ⃗ is the vector shown in Fig. 1 (|κ⃗| = 2π/λ, where λ is the wavelength of the incident field in the medium in which the particle is immersed, and |κ⃗| = |κ⃗0|) and c is the speed of light. The vector r⃗i is assumed to be sufficiently far from the origin (O) that it may be replaced by r⃗ except where it occurs in a phase. The total field at r⃗ , given by

E(s)rt=dE(s)rt,
(4)

is the “scattered” field.

In the laboratory reference frame (x,y,z), the incident electric field propagating in the κ⃗0 direction is given by

E(0)Dit=E(0)exp[i(κ0Diωt)],
(5)

where the field amplitude is resolved into components parallel and perpendicular to the scattering plane (See Fig. 2):

E(0)=E(0)ê(0)+Er(0)êr(0)=(Er(0)E(0)).
Fig. 1. A volume element dVi, located at a point Di from the origin of coordinates (O). The incident radiation is propagating in the κ0 direction and the scattered radiation is propagating in the κ direction. The vector ri is from the volume element dVi, to a distant point at which the scattered field is measured. The vector r is from the origin to the same distant point, which is sufficiently far from the particle that the vectors ri and r are considered to be parallel.
Fig. 2. The plane formed by the propagation vector κ⃗0 of the incident wave and the propagation vector κ⃗ of the scattered wave is the scattering plane. The incident and scattered fields are resolved into components parallel and perpendicular to the scattering plane, i.e., along (ê (0) l , ê (0) r) and (êl, êr), respectively. Θ is the scattering angle.

Resolving the scattered field E⃗ (s) into components parallel and perpendicular to the scattering plane as well (note that ê (0) l and êl, are not parallel), the scattered field at , which is in the form of a spherical wave, can then be written

E(s)=1iκrAE(0)exp[i(κrωt)]or(Er(s)E(s))=1iκr(ArrAlrArlAll)(Er(0)E(0))exp[i(κrωt)],
(6)

where A is the 2×2 scattering amplitude matrix, and

E(s)=E(s)ê+Er(s)êr=(Er(s)E(s)).

3. The differential scattering cross section

To relate the scattered field to scattering cross sections, etc., we recall that the time averaged Poynting vector of the scattered field is

S=κ̂2μ0cEE*+ErEr*=κ̂2μ0cE˜*E=κ̂dPdA,
(7)

Where the superscript * indicates the complex conjugate, the tilde indicates the transposed matrix and, dP is the power crossing an area dA oriented normal to the propagation direction κ⃗ (i.e., the irradiance associated with the propagating field). The differential scattering cross section is defined to be the power scattered into a solid angle dΩ. divided by the irradiance of the incident beam, i.e.,

dσdΩdP(s)dΩdP(0)dA=r2S(s)S(0),
(8)

where the superscript “s” stands for “scattered” and the superscript “0” stands for “incident.” The required Poynting vectors are given by

S(0)=12μ0cE˜(0)*E(0)andS(s)=12μ0c1κ2r2E˜(0)*A˜*AE(0),

so

dσdΩ=1κ2E˜(0)*A˜*AE(0)E˜(0)*E(0).
(9)

If the incident field is unpolarized, then

Er(0)*Er(0)=E(0)*E(0)andEr(0)*E(0)=0=E(0)*Er(0),

and the differential cross section becomes

dσdΩ=1κ2E˜(0)*A˜*AE(0)E˜(0)*E(0)=12κ2(Arr2+Ar2+Ar2+A2).
(10)

(Note: the quantity S 11 defined by Bohren and Huffman [14

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

] is κ2/dΩ.)

4. The Rayleigh-Gans approximation

In its simplest form, in the Rayleigh-Gans Approximation [4

4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).

, 14

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

, 15

15. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

] (RGA) to scattering, the sum in the Eq. (1) is ignored (C = 0), i.e., the only field experienced by dVi is the incident field. Thus, the RGA provides the “zeroth-order” approximation to the scattered field. In the laboratory reference frame (x,y,z), the incident electric field is given by Eq. (5) so the induced dipole moment [Eq. (2)] is

dpi(Di,tric)=ρnα(Di)E(0)(Di,tric)dVi
=ρnα(Di)E(0)exp[i(κ0Diωt+riωc)]dVi.

Then noting that κ = ω/c, we have riω/c = κri = κr-κ⃗●D⃗i,

dpi(Di,tric)=ρnα(Di)E(0)exp[i(κ0κ)Di]exp[i(κrωt)]dVi.
(11)

Resolving the fields into components parallel and perpendicular to the scattering plane, as in Eq. (6) the scattered field is

dE(s)=1iκrdAE(0)exp[i(κrωt)],
(12)

where d A i is the contribution to the matrix A from dVi and is given by

dAi=iρnκ4πε0κ×[κ×α(Di)]exp[i(κ0κ)Di]dVi.
(13)

The scattering plane is the x-z plane (Fig. 2), so

E(0)=E(0)ê0+Er(0)êr0=E(0)êxEr(0)êy,
E(s)=E(s)ê+Er(s)êr=E(s)cosΘêxEr(s)êyE(s)sinΘêz,
κ=κ(êxsinΘ+êzcosΘ),

and

κ0=κêz.

Inserting these into the equation for d A i yields

dAi=iρnκ34πε0(αyyαxyαyzsinΘ+αxycosΘαxzsinΘαxxcosΘ)exp[i(κ0κ)Di]dVi,
(14)

where the α’s are the components of α in the laboratory reference frame (x,y,z). The total scattered field is found by integration over the volume of the object:

A=iρnκ34πε0V(αyyαxyαyzsinΘ+αxycosΘαxzsinΘαxxcosΘ)exp[i(κ0κ)Di]dV.
(15)

If the α’s are independent of position within the particle, the matrix can be removed from the integration. If the particle’s polarizability tensor is isotropic (i.e., αij = α δij) then A reduces to

A=iρnακ34πε0(100cosΘ)Vexp[i(κ0κ)D]dV,
(16)

and the differential cross section becomes

dσdΩ=(ρnακ24πε0)2(1+cos2Θ)2Vexp[i(κ0κ)D]dV2.
(17)

ρnαε0=3(m21m2+2);
(18)

and defining

RVexp[i(κ0κ)D]dV,

we have

dσdΩ=916π2κ4(m21m2+2)2(1+cos2Θ2)R2
(19)

for the differential scattering cross section of a single particle of volume V. If the particle is immersed in a refracting medium, then m is the refractive index of the particle relative to the medium.

The total (σ) and back (σb) scattering cross sections are, respectively,

σ=4πdσdΩdΩandσb=Back2πdσdΩdΩ,
(20)

while the scattering phase and volume scattering functions are

P(Θ)=4πσdσ(Θ)dΩandβ(Θ)=Ndσ(Θ)dΩ,
(21)

where N is the number density of scatterers. The contribution that particles of a given size and shape make to the total scattering coefficient (b) and the backscattering coefficient (bb) are b = and bb = b, respectively. In the RGA, the shape of the particle enters only through the computation of R. Analytic formulas are available for simple shapes, e.g., spheres and cylinders; however, it is easy to carry out the integrations numerically for particles of any shape. For particles other than spheres, R depends on the orientation of the particle. For particles with a given orientational distribution function, dσ/dΩ. must be computed for a large number of orientations and the appropriate weighted average formed.

The fact that dVi is subjected only to the incident field requires that two conditions must hold for the RGA to have validity: (1) there must be insignificant refraction or reflection at the surface of the particle, which implies |m - 1| must be ≪ 1; and (2) the phase of the incident field must not shift significantly over distances of the order of the “size” (L) of the particle, which requires κL|m - 1|≪1.

5. Comparison between RGA and DDA for disk-like particles

In what follows we compare the backscattering cross sections (σb) of randomly-oriented disk-like objects computed via the RGA and the DDA. The DDA results are taken as “exact” computations (the DDA-computed σb’s are expected to be in error by no more than 5%). As an early motivation for such comparisons was interest in the backscattering of coccoliths detached from E. huxleyi suspended in water, I consider disks with diameters 1.5 to 2.75 μm with m = 1.2 (Calcite in water). Figure 3 provides such a comparison for a 2.75 μm homogeneous disk of various thicknesses (t). The comparison shows that the RGA is close to the DDA for t/λ Water less than, or approximately equal to, 0.20 to 0.25. Perhaps more importantly, the comparison shows that σb can be expected to oscillate with increasing t (or decreasing λ Water), i.e., the RGA also provides the qualitative character of the spectral variation of σb. It should be pointed out that the (approximate) “physical optics” developed by model of Gordon and Du [5

5. H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46, 1438–1454 (2001). [CrossRef]

] out performs the RGA when t/λ Water > 0.2, following the DDA reasonably well up to a t/λ Water of 0.8.; however, it cannot be applied to the more complex particle shapes of interest here, e.g., disk-like particles with periodic angular fine structure.

Fig. 3. Comparison of the backscattering cross section computed with the RGA and the DDA for a homogeneous disk of diameter 2.75 μm and thicknesses 0.05, 0.10, and 0.15 μm.

For a more complex example, Fig. 4 compares RGA and DDA computations of σb for the Gordon and Du [5

5. H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46, 1438–1454 (2001). [CrossRef]

] “fishing-reel” model of a detached coccolith. The fishing-reel model consists of two parallel disks of diameter D o with material removed from a concentric circle of diameter Di (i.e., a washer-like object). The two disks are joined together by a hollow cylinder of inner diameter Di and outer diameter Dr. The axis of the cylinder passes through the center of both disks. The individual disks have a thickness of 50 nm and the space between them (the height of the joining cylinder) is t. Table 1 provides values of the parameters of the three fishing-reel models investigated. The three models all have the same volume (∼0.587 μm3). This is accomplished by decreasing the thickness of the wall of the connecting cylinder as shown in Table 1.

For these models, the individual disks have a thickness of 0.05 μm, and therefore are within the t/λ Water < 0.2 criterion from Fig. 3 in the visible. As in Fig. 3, Fig. 4 shows that the RGA and DDA produce qualitatively similar spectral variations, and surprisingly good quantitative agreement even though the total thickness of the particle exceeds λ Water in some cases, and the total diameter is several times λ Water. Comparison of the two suggests that the RGA can be a valuable tool in exploring problems involving multiple disks as long as the individual disks satisfy the t/λ Water < 0.2 criterion.

Fig. 4. Comparison of RGA and DDA computations for the Gordon and Du [5] “fishing reel” models of a detached coccolith.

Table 1:. Parameters of the Gordon and Du [5] “Fishing-reel” model of a detached coccolith.

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View This Table

In an effort to understand the influence of small-scale periodic structure in disk-like objects on backscattering, I examined [11

11. H. R. Gordon, “Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?” Appl. Opt. 45, 7166–7173 (2006). [CrossRef] [PubMed]

] backscattering by a “sectorized” disk formed by starting with a homogeneous disk and removing sectors. Specifically, the disk was divided into equal angle sectors of angle Δα and alternate sectors were removed. The angle Δα was given by

Δα=2π2n

where n is an integer. Figure 5 provides the positions of one layer of dipoles for the resulting structures for n = 4 to 7. I will refer to these objects as “pinwheels.” If we let s be the arc length of the open (or closed) regions at the perimeter of the pinwheel, then s = DdΔα/2, where Dd is the diameter of the disk. The values of s for the various cases that I examined (Dd = 1.5 μm) were such that at a wavelength (λ) of 400 nm in vacuum (300 nm in water), as n progresses from 4 to 7, s took on the values λ, λ/2, λ/4, and λ/8 in water. One of the main goals of my study was to determine if a relationship exists between s and λ where the periodic structure becomes important (or unimportant) to the backscattering.

Fig. 5. Sectorized disks (“pinwheels”) for various values of n used in this study.
Fig. 6. Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.

The results of the computations of the backscattering cross section, σb, carried out for 1.5 μm pinwheels are provided in Fig. 6 (DDA on the left from Ref 11, and RGA on the right), which displays σb, as function of the thickness (t) of the disk divided by the wavelength of the light in water (λ Water). Three thicknesses of the disk are used: 0.05, 0.10, and 0.15 μm. The wavelength λ Water covers the range from 200 nm to over 1000 nm. Note the qualitative similarity between the DDA and the RGA computations. Both show that the backscattering appears to follow a “universal curve” that is close to that for a homogeneous disk with a reduced index m = 1.10 rather than 1.20 (labeled 1.10 in the key to the figure); however, as the wavelength decreases σb, suddenly departs from the universal curve and increases dramatically. This was first observed through extensive computations using the DDA; however, in this case, the behavior could have been predicted based on the RGA computations. (The departure of σb, from the universal curve occurs when the maximum arc length of the open or closed regions of the pinwheel exceeds λ Water/4.). In Fig. 7 the comparisons in Fig. 6 are carried to larger values of t/λ Water, and show that the RGA agrees well with the DDA for values of t/λ Water up to, and somewhat beyond the first maximum that occurs in σb after the departure from the “universal curve.” This maximum is near s/λ Water = 1/2. For larger values of t/λ Water the RGA still provides the qualitative nature of the variation of σb with t/λ Water; however, it no longer quantitatively reproduces the DDA computations.

In Fig. 8 a more complex geometry – parallel pinwheels – is examined. This is somewhat similar to the “fishing reel” model [5

5. H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46, 1438–1454 (2001). [CrossRef]

] for detached coccoliths, but uses sectorized disks (washers) with n = 5 and 6 rather than homogeneous disks and the outside diameter is 1.50 μm rather than 2.75 μm. Again, the agreement between RGA and DDA is quite good throughout the visible, even in a quantitative sense.

Fig. 7. Comparison of DDA and RGA backscattering by sectorized disks in Fig. 5.
Fig. 8. Comparison of DDA and RGA backscattering by parallel sectorized (n = 5 and 6) washers. The individual washers have an outside diameter of 1.50 μm, inside diameter 1.00 μm and thickness 0.05 μm. They are separated by a space of 0.30 μm.

6. Application: estimate of the influence of E. huxleyi birefringence on backscattering

As the E. huxleyi coccolith is composed of calcite, one would expect it to be birefringent. This is indeed the case. The c-axis (optical axis) of the component parts of the E. huxleyi coccolith is radial, i.e., along the “spoke-like” structures [19

19. J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, “Crystal assembly and phylogenetic evolution in heterococcoliths,” Nature 356, 516–518 (1992). [CrossRef]

]. How does this birefringence influence the backscattering? Since RGA provides an adequate description of the backscattering of homogeneous or structured disk-like objects as long as t/λ Water < 0.2, which is satisfied by the individual coccolith plates throughout the visible, we expect that it would apply equally well to a birefringent disk. Thus, we will investigate the possible influence of birefringence on E. huxleyi backscattering by comparing the backscattering in the RGA of a birefringent and an isotropic disk. Computation of the scattering matrix A for an anisotropic disk, for which the optical axis at any point is radial, is sketched out in the Appendix. A uniaxial crystal, Calcite has two refractive indices: me for propagation with the electric vector parallel to the c-axis; and mo for propagation with the electric vector perpendicular to the c-axis. Letting mi represent the refractive index of the isotropic disk, we take

ρnaε0=3(me21me2+2)andρnbε0=3(mo21mo2+2)

for the polarizabilities a and b (see the Appendix) of the birefringent disk, and

ρnαε0=3(mi21mi2+2)

for the isotropic disk. Clearly, mi must depend on mo and me in some manner, and one of the goals of this exercise is to find the combination that provides the best agreement for backscattering of the isotropic and the anisotropic cases. If the disk were composed of small grains of Calcite in random orientation, one would expect [20

20. E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223–2249 (1996). [CrossRef]

] the average refractive index for unpolarized light to be approximately (2mo + me)/3. Using tabulated values for the refractive indices of Calcite [21

21. J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W.G. Driscoll and W. Vaughan, eds., (McGraw-Hill, 1978).

] near 500 nm and taking 1.338 for the index of water, we have mo = 1.241, me = 1.113, and mi = 1.198 (close to the value 1.20 used in the earlier computations). Comparison of the RGA-computed σb, for the radially-anisotropic disk with these values of mo and me with the isotropic disk with index mi = (2mo + me)/3 is provided in Fig. 9. The isotropic disk’s backscattering cross section is seen to be somewhat higher than the birefringent disk. The two can be brought into agreement by taking mi = 1.188 = 0.57 mo + 0.43 me. (For the total scattering cross section, mi = (2mo + me)/3 provides better agreement between the two than mi = 1.188.) This suggests that for the computation of backscattering by model coccoliths with the DDA, a zeroth-order account of the birefringence can be effected by using mi = 1.188 rather than 1.198.

Fig. 9. Comparison of RGA computations of backscattering for a randomly-oriented birefringent disk and an isotropic disk with mi = (2mo + me)/3.

7. Concluding Remarks

The value of the RGA in obtaining qualitative information regarding backscattering by disk-like particles has been demonstrated. The success of the RGA in this case derives from the fact that, unlike a spherical particle with the same mass, most of the volume elements of a thin disk are far enough away from any given element that their interaction is small, i.e., a relatively small amount of the total mass of the particle is close to any one of the volume elements.

I am not advocating the use of the RGA for quantitative computations of σb, for disk-like particles. Rather, because it is computationally fast compared to the DDA, it can be used to explore the backscattering of disk-like models of marine particles for the purpose of either excluding models with unacceptable qualitative behavior, or selecting promising models for further study using the more time-consuming DDA.

8. Appendix: scattering by a birefringent disk

Here we develop the formulas for scattering from a birefringent disk. As our application is to the E. huxleyi coccoliths we take the disk to be uniaxial with the optical axis at any point in the disk in the radial direction. We develop the anisotropic case first and then reduce these formulas to the isotropic case.

A. Anisotropic case

Figure A1 provides the geometry of the scattering problem. The body-fixed coordinate system is cylindrical with radial coordinate ρ the angle η and the coordinate z′ normal to the axis of the disk. In the integral for A, Eq. (15), the required elements of the polarizability matrix (αyy, αxy, αxz, αxy, and αxx) must be provided in laboratory-fixed reference system and depend on the particle’s orientation. However, in the body-fixed reference system the polarizability matrix assumes a particularly simple form:

αB=(αρρ000αηη000αzz)(a000b000b).

The transformation of this matrix to the laboratory-fixed system is straightforward:

α=U˜B˜αBBU,

Fig. A1. A schematic of scattering by a disk. The cylindrical coordinate system (ρ, η, z′) is fixed with respect to the disk (z′ is in the direction of the normal, n).
Fig. A2. Relationship between the laboratory-fixed coordinate system (x, y, z) and the body-fixed system (x′, y′, z′) or (ρ , η , z′). θ, ϕ, and ψ are the Euler angles. Because of the symmetry of the disk the angle ψ can be set to zero.
U=(cosψcosϕcosθsinϕsinψcosψsinϕ+cosθcosϕsinψsinψsinθsinψcosϕcosθsinϕcosψsinψsinϕ+cosθcosϕcosψcosψsinθsinθsinϕsinθcosϕcosθ)

and

B=(cosηsinη0sinηcosη0001).

Because of the symmetry of the disk, the Euler angle ψ is redundant and may be set to zero. The matrix elements of α thus depend on θ, ϕ, and η.

To carry out the required integrations to find A, we need D⃗ = ρ + z, and by resolving (κ⃗0 - κ⃗) into components parallel and normal to the disk’s surface, we find

(κ0κ)D=2κsin(Θ2)[ρcos(ηγ)sinβ+zcosβ]

and

cosβ=cosθsin(Θ2)sinθsinϕcos(Θ2),

where γ is the angle between the component of (κ⃗0 - κ⃗) parallel to the plane of the disk and the x’axis. A typical integral that must be evaluated to find A is then

0t0R02παyyexp{i2κsin(Θ2)[ρcos(ηγ)sinβ+zcosβ]}ρdηdρdz.

Explicit relationships for the components of α are:

αxx=12cos2η[a+b+(ab)cos2ϕ]
+2(a+b)cosηsinηcosθsinϕ
+sin2η[bsin2θ+cos2θ(bcos2ϕ+asin2ϕ)],
αxy=(ab)(cosθcosϕsinη+cosηsinϕ)
×(cosηcosϕcosθsinηsinϕ),
αxz=(ab)sinηsinθ(cosηcosϕcosθsinηsinϕ),
αyy=12cos2η[a+b(ab)cos2ϕ]
2(a+b)cosηsinηcosθcosϕsinϕ,
+sin2η[bsin2θ+cos2θ(acos2ϕ+bsin2ϕ)],
αyz=(ab)sinηsinθ(cosηsinϕ+cosθsinηcosϕ),
αzz=bcos2η+sin2η(asin2θ+bcos2θ),
αyx=αxy,αzx=αxz,αzy=αyz.

Thus, for a given orientation of the disk (given θ and ϕ), all of the required integrals are of the form

0t0R02π(cos2ηcosηsin2ηsinη)exp{i2κsin(Θ2)[ρcos(ηγ)sinβ+zcosβ]}ρdηdρdz.

Figure A3 provides the required η integrals, and this integrates to

2πκ2(1J0(κR)0κRJ1(κR)+J0(κR)1)2ksin(kt2),

where

κ=2κsin(Θ2)sinβ,
k=2κsin(Θ2)cosβ,

and

cosβ=cosθsin(Θ2)sinθsinϕcos(Θ2).

The disk orientation enters through the variation of the αij′s with θ and ϕ as well as through κ and k′.

B. Isotropic case

When the polarizability of the disk material is isotropic, i.e., a = b, then αij = ij, and the scattering amplitude matrix becomes

A=iρnκ3a4πε0(100cosΘ)Vexp[i(κ0κ)D]dV
=iρnκ3a4πε0(100cosΘ)2πκ2κRJ1(κR)2ksin(kt2).

The differential cross section is then

dσdΩ=(ρnκ2a4πε0)2(2VJ1(κR)κRsin(kt2)kt2)2(1+cos2Θ2).

Note that the orientation of the disk enters only through the parameters κ and k′

Fig. A3. The required integrals for the evaluation of the matrix A. The J’s are Bessel functions.

Acknowledgments

References and links

1.

H. R. Gordon and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983). [CrossRef]

2.

D. A. Siegel, A. C. Thomas, and J. Marra, “Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue,” Deep Sea Res. II , 511–3 (2004). [CrossRef]

3.

D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, “The role of seawater constituents in light backscattering in the ocean,” Prog. Oceanogr. 61, 27–56 (2004). [CrossRef]

4.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).

5.

H. R. Gordon and T. Du, “Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi,” Limnol. Oceanogr. 46, 1438–1454 (2001). [CrossRef]

6.

W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, “Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine,” Limnol. Oceanogr. 34, 629–643 (1991). [CrossRef]

7.

W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour, and E. Fernandez, “The 1991 coccolithophore bloom in the central north Atlantic II: Relating optics to coccolith concentration,” Limnol. Oceanogr. 41, 1684–1696 (1996). [CrossRef]

8.

T. J. Smyth, G. F. Moore, S. B. Groom, P. E. Land, and T. Tyrrell, Optical modeling and measurements of a coccolithophore bloom, Appl. Opt. 41, 7679–7688 (2002). [CrossRef]

9.

H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, “Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery,” Geophys. Res. Lett. 28, 1587–1590, (2001). [CrossRef]

10.

W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau, and E. S. Booth, “Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectralradiometer data,” J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560. [CrossRef]

11.

H. R. Gordon, “Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?” Appl. Opt. 45, 7166–7173 (2006). [CrossRef] [PubMed]

12.

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

13.

B. T. Draine and P. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A II, 1491–1499 (1994). [CrossRef]

14.

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

15.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).

16.

L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, “Scattering of light from arbitrarily oriented cylinders,” Appl. Opt. 22, 742–748 (1983). [CrossRef] [PubMed]

17.

K. Shimizu, “Modification of the Rayleigh-Debye approximation,” J. Opt. Soc. Am. 73, 504–507 (1983). [CrossRef]

18.

B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685–697 (1993). [CrossRef]

19.

J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, “Crystal assembly and phylogenetic evolution in heterococcoliths,” Nature 356, 516–518 (1992). [CrossRef]

20.

E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223–2249 (1996). [CrossRef]

21.

J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W.G. Driscoll and W. Vaughan, eds., (McGraw-Hill, 1978).

OCIS Codes
(010.4450) Atmospheric and oceanic optics : Oceanic optics
(290.1350) Scattering : Backscattering

ToC Category:
Atmospheric and ocean optics

History
Original Manuscript: March 20, 2007
Revised Manuscript: April 19, 2007
Manuscript Accepted: April 19, 2007
Published: April 23, 2007

Citation
Howard R. Gordon, "Rayleigh-Gans scattering approximation: surprisingly useful for understanding backscattering from disk-like particles," Opt. Express 15, 5572-5588 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-9-5572


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References

  1. H. R. Gordon and A. Y. Morel, Remote Assessment of Ocean Color for Interpretation of Satellite Visible Imagery: A Review (Springer-Verlag, 1983). [CrossRef]
  2. D. A. Siegel, A. C. Thomas, and J. Marra, "Views of ocean processes from the Sea-viewing wide field-of-view sensor mission: introduction to the first special issue," Deep Sea Res. II,  511-3 (2004). [CrossRef]
  3. D. Stramski, E. Boss, D. Bogucki, and K. J. Voss, "The role of seawater constituents in light backscattering in the ocean," Prog. Oceanogr. 61, 27-56 (2004). [CrossRef]
  4. M. I. Mishchenko, L. D. Travis, and A. A. Lacis (Cambridge, 2002).
  5. H. R. Gordon and T. Du, "Light scattering by nonspherical particles: application to coccoliths detached from Emiliania huxleyi," Limnol. Oceanogr. 46, 1438-1454 (2001). [CrossRef]
  6. W. M. Balch, P. M. Holligan, S. G. Ackleson, and K. J. Voss, "Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine," Limnol. Oceanogr. 34, 629-643 (1991). [CrossRef]
  7. W. M. Balch, K. Kilpatrick, P. M. Holligan, D. Harbour and E. Fernandez, "The 1991 coccolithophore bloom in the central north Atlantic II: Relating optics to coccolith concentration," Limnol. Oceanogr. 41, 1684-1696 (1996). [CrossRef]
  8. T. J. Smyth, G. F. Moore, S. B. Groom, P. E. Land and T. Tyrrell, Optical modeling and measurements of a coccolithophore bloom, Appl. Opt. 41, 7679-7688 (2002). [CrossRef]
  9. H. R. Gordon, G. C. Boynton, W. M. Balch, S. B. Groom, D. S. Harbour, and T. J. Smyth, "Retrieval of Coccolithophore Calcite Concentration from SeaWiFS Imagery," Geophys. Res. Lett. 28, 1587-1590, (2001). [CrossRef]
  10. W. M. Balch, H. R. Gordon, B. C. Bowler, D. T. Drapeau and E. S. Booth, "Calcium carbonate measurements in the surface global ocean based on Moderate-Resolution Imaging Spectralradiometer data," J. Geophys. Res. 110C, C07001 (2005), doi:l0.1029j2004JC002560. [CrossRef]
  11. H. R. Gordon, "Backscattering of light from disk-like particles: is fine-scale structure or gross morphology more important?" Appl. Opt. 45, 7166-7173 (2006). [CrossRef] [PubMed]
  12. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333,848-872 (1988). [CrossRef]
  13. B. T. Draine and P. Flatau, "Discrete-dipole approximation for scattering calculations," J. Opt. Soc. Am. A ll,1491-1499 (1994). [CrossRef]
  14. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).
  15. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, 1957).
  16. L. D. Cohen, R. D. Haracz, A. Cohen, and C. Acquista, "Scattering of light from arbitrarily oriented cylinders," Appl. Opt. 22, 742-748 (1983). [CrossRef] [PubMed]
  17. K. Shimizu, "Modification of the Rayleigh-Debye approximation," J. Opt. Soc. Am. 73, 504-507 (1983). [CrossRef]
  18. B. T. Draine and J. Goodman, Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation, Astrophys. J. 405, 685-697 (1993). [CrossRef]
  19. J. R. Young, J. M. Didymus, P. R. Brown, B. Prins, and S. Mann, "Crystal assembly and phylogenetic evolution in heterococcoliths," Nature 356, 516-518 (1992). [CrossRef]
  20. E. Aas, Refractive index of phytoplankton derived from its metabolite composition, J. Plankton Res. 18, 2223-2249 (1996). [CrossRef]
  21. J. M. Bennett and H. E. Bennett, "Polarization," in Handbook of Optics, W.G. Driscoll and W. Vaughan, eds., (McGraw-Hill, 1978).

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