Polarization of the underwater light field • luxR

Sunlight scattered by water becomes partially linearly polarized, and many aquatic animals — cephalopods, stomatopods, many crustaceans and some fish — see that polarization and use it for navigation, contrast enhancement, and communication. luxR provides a tunable, Rayleigh-like single-scattering approximation to the underwater polarization field: enough to build intuition and explore geometry, but not a vector radiative-transfer solution.

Model status: underwater_polarization() is an uncalibrated approximation, not a quantitative prediction. Its result carries a luxR.polarization attribute recording that status and the exact parameter values. Quantitative use requires calibration for the relevant wavelength, water type, surface state, depth, and viewing geometry.

The model

For singly (Rayleigh-like) scattered light, the degree of polarization (DoP) depends on the scattering angle $\Theta$ between the line of sight and the refracted solar beam,

$DoP(\Theta) = DoP_{max}\,\frac{\sin^2\Theta}{1 + \cos^2\Theta},$

peaking at $\Theta = 90^\circ$ and vanishing toward the sun and anti-sun. With depth, multiple scattering randomizes the field, so DoP relaxes toward an asymptotic floor. The sun’s beam is refracted at the surface into Snell’s window first:

refracted_solar_angle(c(0, 30, 60, 90))   # in-water solar zenith (deg)

## [1]  0.00000 22.03011 40.51759 48.60663

Degree of polarization across the sky

With the sun overhead, polarization is strongest when looking horizontally (90° from the sun) and disappears looking straight up at the sun:

vz  <- seq(0, 90, by = 2)
res <- underwater_polarization(view_zenith = vz, sun_zenith = 0, depth = 1)

plot(res$scattering_angle, res$dop, type = "l", lwd = 2, col = "steelblue4",
     xlab = "Scattering angle from sun (deg)",
     ylab = "Degree of polarization",
     main = "Underwater DoP vs angle from the sun (near surface)")
abline(v = 90, lty = 3, col = "grey50")

How polarization fades with depth

Looking at the 90° band (maximally polarized at the surface), DoP relaxes toward the deep asymptotic value as multiple scattering takes over:

depths <- seq(0, 100, by = 2)
dz <- underwater_polarization(view_zenith = 90, sun_zenith = 0,
                              depth = depths, dop_max = 0.4, dop_asym = 0.03)

plot(depths, dz$dop, type = "l", lwd = 2, col = "firebrick",
     xlab = "Depth (m)", ylab = "Degree of polarization",
     main = "DoP at the 90-degree band vs depth")
abline(h = 0.03, lty = 3, col = "grey50")

The dop_max, dop_asym, and z_p arguments let you tune the approximation to a water type or to field measurements. Their defaults are illustrative:

Parameter	Default	Evidence and valid domain
`dop_max`	0.4	Upper end of the roughly 0.25–0.40 maxima reported by Cronin & Shashar (2001) at 15 m in clear tropical marine water. The function defines this parameter at depth zero, so the observation is context rather than a calibration.
`dop_asym`	0.03	Heuristic deep-field floor. It has no empirical universal domain and must be calibrated for quantitative use.
`z_p`	20 m	Heuristic relaxation scale, not a measured attenuation length. It must be calibrated for quantitative use.
`n`	1.333	Conventional visible-light approximation for water. Refractive index varies with wavelength, temperature, and salinity.

Cronin & Shashar measured clear tropical reef water at 15 m over 350–600 nm and found the strongest polarization 60–90° from the sun. The package tests this angular pattern as a literature-consistency benchmark. It does not treat that qualitative agreement as calibration.

The e-vector

The angle of polarization (aop) is perpendicular to the scattering plane. It is returned alongside DoP and is NA when looking straight along the solar beam, where the e-vector is undefined:

underwater_polarization(view_zenith = c(0, 45, 90), view_azimuth = 0,
                        sun_zenith = 30, sun_azimuth = 0, depth = 5)

##   scattering_angle        dop aop
## 1         22.03011 0.03020901  90
## 2         22.96989 0.03231232  90
## 3         67.96989 0.24130938  90

For Stokes-vector inputs you can also compute the descriptors directly with degree_of_polarization() and angle_of_polarization(). Invalid Stokes states whose polarized intensity exceeds total intensity are rejected rather than silently returning a degree greater than one.