Page updated: April 16, 2020
Author: Curtis Mobley

Fresnel Equations for Polarization

This page shows how polarizied light is reflected and transmitted by a level air-water surface. The geometry is the same as for the Level 1 discussion of Fresnel reflectance and transmittance of unpolarized light by a level sea surface. Now, however, the state of polarization of the incident light is described by a four-component Stokes vector, as described on the page on the Vector RTE. Consequently, reflection and transmission by the surface are described by 4 × 4 matrices.

The state of polarization of a light field is specified by the four-component Stokes vector, whose elements are related to the complex amplitudes of the electric field vector E resolved into directions that are parallel (E) and perpendicular (E) to a conveniently chosen reference plane. However, there are two versions of the Stokes vector seen in the literature, and these two versions have different units and refer to different physical quantities. The coherent Stokes vector describes a quasi-monochromatic plane wave propagating in one exact direction, and the vector components have units of power per unit area (i.e., irradiance) on a small surface element perpendicular to the direction of propagation. The diffuse Stokes vector describes light propagating in a small set of directions surrounding a particular direction and has units of power per unit area per unit solid angle (i.e., radiance). It is the diffuse Stokes vector that appears in the vector radiative transfer equation. The differences in coherent and diffuse Stokes vectors are rigorously discussed in (Mischenko, 2008).

For either air- or water-incident light, S̲i denotes the diffuse Stokes vector of the incident light, S̲r is the reflected light, and S̲t is the transmitted light. Angles 𝜃i, 𝜃r, and 𝜃t are the incident, reflected, and transmitted directions of the light propagation measured relative to the normal to the surface. For a level surface, S̲i, S̲r, and S̲t all lie in the same plane.

There are four matrices to describe reflection and transmission: R̲aw describes how air-incident light is reflected by the water surface back to the air, T̲aw describes how air-incident light is transmitted through the surface into the water, R̲wa reflects water-incident light back to the water, and T̲wa transmits light from the water into the air. However, because S̲i, S̲r, and S̲t are coplanar, scattering by the level surface does not involve rotation matrices as does scattering within the water body. (Or, from another viewpoint, the incident and final meridian planes and the scattering plane are all the same, the rotation angles between meridian and scattering planes are 0, and the rotation matrices reduce to identity matrices.)

The reflection and (especially) transmission of polarized light by a dielectric surface such as a level water surface are rather complicated processes, and the literature contains a number of different (and, indeed, sometimes incorrect) mathematical formulations of the equations. The formulas given in (Garcia, 2012) are used here. Note, however, that although the equations in (Garcia, 2012) are correct, some of his derivations and interpretations are incorrect, as explained by (Zhai, et al. (2012). Both papers must be used to understand the equations now presented. The equations in Garcia will be referenced by (G21) and so on; the corresponding equations in (Zhai et al. (2012) will be referenced as (Z5), etc.

The reflectance and transmittance matrices have a general formulation for the interface between any two dielectric media a and b. Let na be the index of refraction of medium a and nb be that of medium b. In general na and nb are complex numbers, but for the air-water surface we take nair = 1 and nwater 1.34 to be real indices of refraction. For reflection, the reflected angle 𝜃r equals the incident angle 𝜃i. For transmission from a to b, the transmitted angle is given by Snel’s law, na sin𝜃a = nb sin𝜃b, or

𝜃b = arcsin na sin𝜃i nb . (1)

For water-incident light, na = nwater and nb = nair, in which case the transmitted angle becomes undefined beyond the critical angle for total internal reflection, which for water is 𝜃c = arcsin(1nwater) 48deg. For water-incident angles greater than 𝜃c the incident light is totally reflected back to the water and no light is transmitted to the air.

Let R̲ab denote the reflectance matrix for light incident from medium a and reflected back by medium b. R̲ab thus represents either R̲aw or R̲wa. Likewise, let T̲ab denote the reflectance matrix for light incident from medium a and transmitted through the surface into medium b. T̲ab thus represents either T̲aw or T̲wa.

With these preliminaries, the reflectance matrix R̲ab is (G10)

R̲ab = 1 2(RR + RR)1 2(RR RR) 0 0 1 2(RR RR)1 2(RR + RR) 0 0 0 0 {RR} {RR} 0 0 {RR}{RR} . (2)

Here {RR} denotes the real part of RR and {RR} is the imaginary part.

The transmission matrix T̲ab is (G11 or Z3)

T̲ab = fT 1 2(TT + TR)1 2(TT TT) 0 0 1 2(TT TR)1 2(TT + TT) 0 0 0 0 {TT} {TT} 0 0 {TT}{TT} . (3)

The components of these equations are given by (G7):

R = nb cos𝜃a na cos𝜃b nb cos𝜃a + na cos𝜃b (4) R = na cos𝜃a nb cos𝜃b na cos𝜃a + nb cos𝜃b (5) T = 2na cos𝜃a nb cos𝜃a + na cos𝜃b (6) T = 2na cos𝜃a na cos𝜃a + nb cos𝜃b. (7) The factor fT is defined below in Eq. (23). In general, the indices of refraction are complex numbers and these equations must be used. However, for real indices of refraction, the matrix elements can be simplified at the expense of having a special case for water-incident angles greater that the critical angle.

Define

nab = na nbandnba = nb na. (8)

Then for the case of air-incident light, i.e., na nb, or water-incident light with the incident angle less than the critical angle, i.e., na > nb and 𝜃a < 𝜃c, the equations yield the real forms (G14 and G15)

RR = cos𝜃a nab cos𝜃b cos𝜃a + nab cos𝜃b 2 (9) RR = nab cos𝜃a cos𝜃b nab cos𝜃a + cos𝜃b 2 (10) {RR} = cos𝜃a nab cos𝜃b cos𝜃a + nab cos𝜃b nab cos𝜃a cos𝜃b nab cos𝜃a + cos𝜃b (11) {RR} = 0 (12) TT = 2nab cos𝜃a cos𝜃a + nab cos𝜃b 2 (13) TT = 2nab cos𝜃a nab cos𝜃a + cos𝜃b 2 (14) {TT} = 4nab2 cos2𝜃a (cos𝜃a + nab cos𝜃b)(nab cos𝜃a + cos𝜃b) (15) {TT} = 0. (16)

It should be noted that for the case of normal incidence, 𝜃i = 0, both RR and RR reduce to

RR = R R = nb na nb + na 2. (17)

This gives a reflectance of Rab(𝜃i = 0) = 0.021 for nwater = 1.34, for both air- and water-incident light.

For the case of total internal reflection, i.e., na > nb and 𝜃a 𝜃c, the following equations are to be used (G22):

RR = 1 (18) RR = 1 (19) {RR} = 2sin4𝜃a 1 (1 + nba2)cos2𝜃a 1 (20) {RR} = 2cos𝜃a sin2𝜃asin 2 𝜃a nba 2 1 (1 + nba2)cos2𝜃a (21)

and all elements of the transmission matrix elements are 0:

T̲ab = T̲wa = 0̲. (22)

Finally, the all-important transmission factor fT in Eq. (3) is given by

fT = nba3 cos𝜃b cos𝜃a (fordiffuseStokesvectors), (23)

when computing the transmittance for diffuse Stokes vectors. These equations give everything needed to describe reflection and transmission of polarized light by a level sea surface.

Figure 1 shows the R̲aw and T̲aw matrices as a function of incident angle 𝜃i for nair = 1 and nwater = 1.34. The (1,1) matrix elements are shown in the upper-left plot, and the (4,4) elements are in the lower-right plot. The red curves are R̲aw(𝜃i) and the blue curves are T̲aw(𝜃i). The reflectance curve for Raw(1,1) is the Fresnel reflectance for unpolarzed light as given in the section on Fresnel formulas for unpolarized light: it starts at 0.021 for normal incidence and nwater = 1.34, and rises to 1 at grazing incidence. The transmission curve for Taw(1,1) on the other hand may look incorrect because it has values greater than one. Its maximum value at normal incidence is

Taw(1,1) = 4nb3 (1 + nb)2 = 1.758 (24)

However, this value is indeed correct and is a consequence of the fact that we are now dealing with a diffuse Stokes vector with units of radiance, and the n2 law for radiance applies. The curves in Fig.(1) agree exactly with the corresponding plots in (Garcia, 2012) (his Figs. 1-3).


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Figure 1: Reflectance and transmittance matrices as functions of the incident angle 𝜃i for air-incident radiance. R̲aw is in red and T̲aw is in blue. The vertical dotted line at 𝜃i = 53.3deg is Brewster’s angle.

If we were dealing with coherent Stokes vectors with units of irradiance, then the fT factor of Eq. (23) would be

fT = nba cos𝜃b cos𝜃a(forcoherentStokesvectors). (25)

The transmittance for normal incidence then would be (4nb)(1 + nb)2 = 0.979, which with the reflectance sums to one (and also sums to one for all other incident angles). As noted elsewhere, it is the Law of Conservation of Energy, not the law of conservation of radiance.

The vertical dotted line in Fig. (1) shows the location of Brewster’s angle,

𝜃Brew = arctan(nb) (26)

which is arctan(1.34) = 53.3deg in the present case. At this angle, Raw(1,2) = Raw(2,1) = Raw(1,1), and Raw(3,3) = Raw(4,4) = 0. In the present case Raw(1,1) 0.04 at 𝜃Brew, and the reflection process S̲r = R̲aw(𝜃i = 𝜃Brew)S̲i becomes

S̲r = 0.04 0.04000.04 0.04 0 0 0 0 00 0 0 0 0 I 0 0 0 = 0.04I 0.04I 0 0 . (27)

Thus, at Brewster’s angle, unpolarized incident radiance is totally horizontally polarized upon reflection.

It should also be noted that the non-zero Taw(2,1) means that unpolarized radiance becomes partly horizontally polarized upon transmission through the surface.

Figure (2) shows R̲aw and T̲aw as reduced scattering matrices, i.e. after dividing each element by its (1,1) component. These plots show more clearly the behavior of the R̲aw matrix elements at Brewster’s angle. These curves agree exactly with the corresponding plots in (Kattawar and Adams (1989) (their Fig. 4).


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Figure 2: Reduced reflectance and transmittance matrices for air-incident radiance [the reflectance and transmittance matrices of Fig.(1) normalized by their (1,1) elements]. The vertical dotted line is Brewster’s angle.

Figure (3) shows R̲wa and T̲wa. The vertical dotted line is at the critical angle for total internal reflection, which in the present case is 𝜃c = 48.3deg. For angles less than the critical angle, the transmission is never more than about 0.54. This again shows the n-squared law for radiance. In going from water to air, the in-water radiance is decreased by a factor of 1nwater2 when crossing the surface because the solid angle in air is greater than that in water by a factor of nwater2. The (1,1) elements show that beyond the critical angle there is no transmission and total reflection. These curves agree with the corresponding plots in (Garcia (2012) (his Figs. 4-6).

Figure (4) shows the reduced water-to-air matrices. These curves agree with the corresponding plots in (Kattawar and Adams (1989) (their Fig. 5. The signs of the T̲wa(3,4) and T̲wa(4,3) elements are reversed in the original Fig. 5, which had a sign error.).


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Figure 3: Reflectance and transmittance matrices as functions of the incident angle 𝜃i for water-incident radiance. R̲wa is in red and T̲wa is in blue. The vertical dotted line is the critical angle for total internal reflectance.


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Figure 4: Reduced reflectance and transmittance matrices for air-incident radiance [the reflectance and transmittance matrices of Fig.(3) normalized by their (1,1) elements]. The 34 and 43 elements are the reverse of Fig. 5 in (Kattawar and Adams (1989) due to a sign error in the original paper.

The non-zero matrix elements of course depend on incident angle as seen above, but also depend weakly on the wavelength via the wavelength dependence of nwater.

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