H. Neuberger, Introduction to Physical Meteorology,
Revised ed. (Pennsylvania State University, University Park, Pa., 1957), pp.
Reference 1, pp. 269–270, lists skylight
polarization studies by several of Rayleigh’s
K. L. Coulson, Polarization and Intensity of Light
in the Atmosphere (Deepak, Hampton, Va., 1988), pp. 375–391.
See Ref. 1, pp. 377–378.
E. J. McCartney, Optics of the Atmosphere:
Scattering by Molecules and Particles (Wiley, New York, 1976), pp. 213,
For examples, see Ref. 1, pp.
Ref. 5, p. 197.
Ref. 1, p. 233. Also see E. Collett , Polarized
Light: Fundamentals and Applications (Marcel Dekker, New York, 1993), pp.
C. F. Bohren, D. R. Huffman, Absorption and
Scattering of Light by Small Particles (Wiley, New York, 1983), pp.
Ref. 20, p. 53.
See Ref. 1, p. 254 and Ref. 20, p.
Ref. 20, pp. 46, 50. χ has the same direction as
skylight’s plane of polarization but avoids the conceptual difficulties that
a plane of (partial) polarization entails.
For examples, see Ref. 1, pp. 554–555 and
For example, see Ref. 5, pp. 194–197. Lines of zero
Q are still called neutral lines (Ref. 1, pp. 254–258).
ϕrel ranges between 0° and 360°, with values
increasing clockwise from the Sun’s azimuth.
Ref. 1, p. 254.
Ref. 20, pp. 382–383. Equation (4) also defines
polarization for specular reflection from planar surfaces. χ is horizontal
for linear polarization by reflection from horizontal surfaces (e.g., calm
water). To measure this polarization, once again set Eq. (4)’s 0° direction
parallel to χ (i.e., horizontal).
Ref. 20, p. 54.
Ref. 14, pp. 198–199.
Ref. 14, pp. 136–139. Note that 475 nm is a
dominant wavelength typical of clear skies.
See Ref. 1, pp. 522–525, for a discussion of
partial polarization on reflection by water.
Ref. 1, p. 311 (Fig. 5.22). Large near-horizon pQ
gradients at 90° from a low Sun appear consistently in my polarization
Usually red pixels in Figs. 2–4 are the result of
identical 24-bit colors in the original digital images; so, in a limited
sense, the maps do include points where pQ and P = 0.0 exactly, but this
equality is just an artifact of the resolution with which the slide scanner
quantized scene radiances.
For a remote-sensing application of P derived from
photographs, see K. L. Coulson, V. S. Whitehead, C. Campbell, “Polarized
views of the earth from orbital altitude,” in Ocean Optics VIII, M. A.
Blizard, ed., Proc. SPIE637, 35–41 (1986). Narrow-FOV photographic
polarimetry that uses a Savart plate is discussed in R. Gerharz,
“Polarization of scattered horizon light in inclement weather,” Arch.
Meteorol. Geophys. Bioklimatol. Ser. A 26, 265–273 (1977).
For example, see Ref. 1, p. 261 (Fig. 4.36). θv is
0° at the astronomical horizon, except in Figs. 10–13, where θv is 0° at the
slightly higher mean topographic horizon (see Fig. 3).
My polarizer’s H90 is fairly uniform at visible
wavelengths, although crossed pairs of such polarizers do transmit a dim
violet from a white-light source. Because skylight dominant wavelengths at
the Earth’s surface typically are 475 nm or more, the increase in
photographic polarizers’ H90 at shorter wavelengths is unlikely to
appreciably bias observations of skylight polarization.
See Ref. 1, p. 582, for the general form of these
Mueller matrix calculations.
As noted above, P measured by the four-image
technique depends only on a polarizer’s relative (rather than absolute)
directions of 0°, 45°, 90°, and 135°. In other words, the four-image 0°
direction can differ arbitrarily from χ.
See Ref. 1, pp. 254–258.
Ref. 20, pp. 112–113.
Ref. 1, pp. 391–393.