Page updated: April 11, 2020
Author: Emmanuel Boss
View PDF

# Size Effect on Modeled Optical Properties

Size has a major inﬂuence on the optical properties of particles as well as on other particle properties (mass, settling rates, carbon content etc’). Size is therefore an important input when computing the optical properties of marine particles as well as a parameter we use optical properties to invert for. It is often not obvious how to assign a single size to describe a non-spherical marine particle. e.g. both surface area and volume have important optical, biogeochemical and physical consequences.

Size eﬀects on the optical properties depends on the ratio of the particle size (e.g. its diameter D)to the wavelength of light in the medium $x=\frac{\pi D}{{\lambda }_{m}}$ and the diﬀerence of index of refraction between medium (the reference relative to which we compute the index of refraction of the particle, $m=n+ik$) and particle $|n-1|$; The diﬀerence in time it takes a ray to pass through the particle compared to the same distance in the free medium is proportional to the product $x|n-1|$.

#### 0.1 particle size $\ll$ wavelength

When particles are much smaller than the wavelength (Rayleigh’s regime), $D\ll {\lambda }_{m}$, the optical properties of a particle are independent of shape, but dependent on the particle’s volume (e.g. van de Hulst, 1974, Bohren and Huﬀman, 1983, Kokhanovsky, 2004).

The scattering and absorption cross-sections (equivalent to the scattering and absorption of a single particle per ${m}^{3}$) is given by:

 ${C}_{b}=\frac{24{\pi }^{3}{V}^{2}}{{\lambda }^{4}}{|\frac{{m}^{2}-1}{{m}^{2}+2}|}^{2}\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{2}\right].$ (1)
 ${C}_{a}=\frac{6\pi V}{\lambda }Im\left(\frac{{m}^{2}-1}{{m}^{2}+2}\right)\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{2}\right].$ (2)

where $V$ the particle’s volume. It follows that the volume speciﬁc absorption is independent of size while the volume speciﬁc scattering increases linearly with the particle’s volume (e.g. as ${D}^{3}$). Similarly the spectral behavior is of the resulting attenuation is very diﬀerent dependent on whether it is dominated by absorption or scattering. It will vary as ${\lambda }^{-1}$ if dominated by absorption or on ${\lambda }^{-4}$ if dominated by scattering. The phase function in that case is symmetric and is given by:

 $\stackrel{̃}{\beta }=\frac{3\left[1+co{s}^{2}\left(𝜃\right)\right]}{4}\phantom{\rule{3.04076pt}{0ex}}\left[S{r}^{-1}\right].$ (3)

#### 0.2 particle size $\gg$ wavelength

At the other side of the size spectrum $x\gg {\lambda }_{m}$. This is the geometric optics regime. If the particles are absorbing even little and no light makes it through the particle,

 ${C}_{a}={C}_{b}=\left[{m}^{2}\right].$ (4)

Where $$ is the average (over all orientation) cross-sectional area. For a randomly oriented convex particle $$ equals a forth of the surface area. It follows that absorption and scattering are insensitive to wavelength and that per volume (or mass) they decrease with size as $\frac{1}{D}$. In other words, the larger the particle the less of its material interacts with light. The phase function is not a available in close form for all angles and is more sharply peaked in the forward direction. The larger the particle the more it is peaked.

#### 0.3 ’Soft’ particles (Anomalous diﬀraction)

For particles that have an index of refraction close to that of the medium (’soft’ particles) analytical solutions have been found (van de Hulst, 1957) for the range spanning between Rayleigh’s solution and Geometric optics.

 ${C}_{a}=\frac{\pi {D}^{2}}{2}\left(\frac{1}{2}+\frac{exp\left(-4kx\right)}{4kx}+\frac{exp\left(-4kx\right)-1}{{\left(4kx\right)}^{2}}\right)\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{2}\right].$ (5)
 ${C}_{ext}=\pi {D}^{2}\left(\frac{1}{2}-\frac{{e}^{\left(-\rho tan\beta \right)}cos\beta }{\rho }sin\left(\rho -\beta \right)-{\left(\frac{cos\beta }{\rho }\right)}^{2}cos\left(\rho -2\beta \right)+{\left(\frac{cos\beta }{\rho }\right)}^{2}cos\left(2\beta \right)\right)\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{2}\right].$ (6)

#### 0.4 Mie’s solution (Mie theory)

Mie (1908) provided a solution for light interaction (absorption, scattering, attenuation and polarization) with a homogeneous spheres of arbitrary size and index of refraction. This solution has been used extensively in ocean optics to derive insight into light interaction with marine particles despite the fact that marine particles are not homogeneous nor spheres. It is, however, a very good ﬁrst step in the direction of understanding how marine particles interact with light. Mie’s result depends on the following particle properties: its size relative to the wavelength and its (complex) index of refraction relative to the medium in which it is immersed.

Assigning an index of refraction to a (non-homogeneous) marine particle is not obvious. Aas (1996) discuss in details how the ’average’ index of refraction of phytoplankton depends on the optical properties of the constituents of such a cell. One of the methods to obtain the real part of the index of refraction is to immerse the particles in a series of oils each with its own index of refraction and ﬁnd the one in which the particle exhibits the minimum scattering.

Once a size and index of refraction (relative to the medium) are chosen, we seek a solution of Maxwell’s equation in a non-absorbing medium. A plane monochromatic wave excites the sphere which than radiates waves in space (scattering). Some of the light is attenuated within the particles (absorption). The solution we seek is the sum of the monochromatic wave and the scattered wave. The spherical symmetry of the problem facilitate its treatment (for a full exposition see Ch. 4 in Bohren and Hufman, 1983).

The output of a Mie code, such as the one we attach here (a Matlab translation of that in the appendix of Bohren and Huﬀman, 1983), consists of the optical eﬃciency factors for attenuation and scattering (${Q}_{b}$ and ${Q}_{ext}$, that for absorption is obtained by diﬀerence) and two complex angular scattering vectors linking scattered and incident ﬁelds (${S}_{1}$ and ${S}_{2}$) from which the phase function $\stackrel{̃}{\beta }$ can be obtained (as well as all other elements of the Mueller scattering matrix). The optical eﬃciency factors are the ratio of optical cross-sections to the geometrical cross-sections.

#### 0.5 Computing the optical properties of a suspension of particles

For a suspension of particles with $n$ particles $\left[$${m}^{-3}\right]$ all with the same physical and optical properties the absorption, scattering attenuation and VSF are simply:

 $\left(a,b,c\right)=n\left({C}_{a},{C}_{b},{C}_{ext}\right)=n\left({Q}_{a},{Q}_{b},{Q}_{ext}\right)\frac{\pi {D}^{2}}{4}\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}\right].$ (7)
 $\beta =n{C}_{b}\stackrel{̃}{\beta }\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}S{r}^{-1}\right].$ (8)

If rather than a single sized suspension, we have a suspension with a particulate size distribution, $f\left(D\right)$, such that:

 $N\left(\Delta D={D}_{2}-{D}_{1}=\underset{{D}_{1}}{\overset{{D}_{2}}{\int }}f\left(D\right)dD\phantom{\rule{3.04076pt}{0ex}}\left[\text{#}{m}^{-3}\right],$ (9)

the IOPs are computed from:

 $\left(a,b,c\right)=\underset{{D}_{min}}{\overset{{D}_{max}}{\int }}{C}_{a,b,c}\left(D\right)f\left(D\right)dD\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}\right],$ (10)
 $\beta =\underset{{D}_{min}}{\overset{{D}_{max}}{\int }}\stackrel{̃}{\beta }\left(D\right){C}_{b}\left(D\right)f\left(D\right)dD\phantom{\rule{1em}{0ex}}\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}S{r}^{-1}\right].$ (11)

In reality, we do not get from our instruments (e.g. Coulter counter, LISST) continuous size distributions but rather piecewise constant PSDs:

 (12)

Where ${N}_{i}$ denotes the number of particles within a bin size within the volume sampled, e.g. with units of $\left[$#${m}^{-3}\right]$. The bins boundaries are usually speciﬁed such that each successive bin is larger than the previous by a constant factor, e.g. ${D}_{m}=\delta {D}_{m-1}={\delta }^{m}{D}_{min}$. The size assigned to the bin is commonly the geometric average of its boundaries, e.g. $D\left(m\right)=\sqrt{{D}_{m}{D}_{m-1}}$ (as opposed to the mean size). If the bins are suﬃciently small such that the cross-sections do not vary much within them, the IOP are computed as follows:

 $\left(a,b,c\right)=\sum _{m=1}^{m=n}{C}_{a,b,c}\left({D}_{m}\right){N}_{m}\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}\right],$ (13)
 $\beta =\sum _{m=1}^{m=n}\stackrel{̃}{\beta }\left({D}_{m}\right){C}_{b}\left({D}_{m}\right){N}_{m}\phantom{\rule{3.04076pt}{0ex}}\left[{m}^{-1}S{r}^{-1}\right].$ (14)