Page updated: April 15, 2020
Author: Curtis Mobley

The Level Sea Surface

This chapter discusses how light is reflected by and transmitted through air-water surfaces, and how it is reflected by opaque surfaces such as the ocean bottom. The Level 1 discussion begins with the basics of reflection and transmission of unpolarized light by a level or flat water surface. Although the ocean is rarely glassy smooth, reflection and transmission by rough, wind-blown surfaces are modeled using the equations for a flat surface applied to each small patch of sea surface, which although tilted from the normal to the mean sea surface can be assumed to be locally flat. Other Level 1 material introduces the bidirectional reflectance distribution function or BRDF. The BRDF is the fundamental quantity for specifying how an opaque surface reflects light.

The Level 2 material first considers reflection and transmission for polarized light. Then several pages show how wind-blown sea surfaces can be described in terms of energy spectra and, conversely, how random sea surfaces can be generated starting with energy spectra. These techniques are widely used both for generation of sea surfaces for quantitative modeling of reflection and transmission, and for computer animation of sea surfaces as used in many movies.

Geometric Relations

The wavelength of visible light is much, much less than the millimeter and larger spatial wavelengths of the waves on wind-blown surfaces. Therefore, the laws of geometrical optics and the idealization of a narrow ray of collimated light give a good description of the relevant physical processes.

Figure 1 illustrates a level surface with light incident onto the surface from the air side (panel a), and from the water side (panel b). The real index of refraction of the air is na, which is taken to be one. nw is the real index of refraction of the water, which is approximately 1.34 at visible wavelengths. n̂ is a unit vector normal to the surface. ξ̂i is a unit vector in the incident direction; ξ̂r and ξ̂t are respectively the directions of the reflected and transmitted rays. 𝜃i = cos1(|ξ̂i n̂|) is the acute angle between the incident direction and the normal, and 𝜃r and 𝜃t are the angles of the reflected and transmitted rays relative to the normal.


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Figure 1: Illustration of incident, reflected, and transmitted rays for air- and water-incident light.

The incident, reflected, and refracted directions all lie in the plane defined by ξ̂ i and n̂. The reflected angle is always equal to the incident angle: 𝜃r = 𝜃i, which is known as the Law of Reflection. The incident and transmitted angles are related by

n1 sin𝜃1 = n2 sin𝜃2, (1)

where subscripts 1 and 2 refer to any two media. This equation is usually called Snell’s law, although more properly it should be Snel’s law. It was rediscovered in the west by Willebrord Snel van Royen (1580-1626). In the days when European scientists published in Latin, Snel’s name was Latinized to Snellius, which became Snell as Latin was replaced by German and then English as the common language of physical science. However, this law (in a different but equivalent form) can be traced back to the treatise On Burning Mirrors and Lenses published by the Persian Abu ibn Sahl in Bagdad in 984.

Figure 2 gives a visual representation of the relations between the various unit vectors, angles, and indices of refraction.


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Figure 2: Graphical representations of relations among incident, reflected, and transmitted directions. Reproduced from Fig. 4.2 of Light and Water (1994), where ξ̂ i = ξ̂.

For air-incident light, na = 1 and Snel’s law reads sin𝜃i = nw sin𝜃t. Then the angle of transmission is given by

𝜃t = sin1 1 nw sin𝜃i . (2)

The relations between the unit vectors are given by the following equations (with na = 1):

ξ̂r = ξ̂i 2(ξ̂i n̂)n̂,

and

ξ̂t = 1 nw(ξ̂i cn̂),

where

c = ξ̂i n̂ + (ξ ̂ i n ̂ )2 + nw 2 1.

For the water-incident case, Snel’s law reads nw sin𝜃i = sin𝜃t, in which case the angle of transmission is given by

𝜃t = sin1(n w sin𝜃i). (3)

The relations between the unit vectors are then

ξ̂r = ξ̂i 2(ξ̂i n̂)n̂,

and

ξ̂t = nwξ̂i cn̂,

where

c = nwξ̂i n̂ (nw ξ ̂ i n ̂ )2 nw 2 + 1.

If nw sin𝜃i < 1, Eq. (3) gives a real value for 𝜃t and light is transmitted from the water to the air. However, if 𝜃i is greater than the critical angle for total internal reflection

𝜃c = sin1(1n w), (4)

then there is no real solution for the inverse sine. In this case, all light incident onto the water side of the air-water surface is reflected back into the water. This is called total internal reflection. The dotted line in the right panel of Fig. 1 represents the critical angle. The red unit vectors illustrate the case 𝜃i < 𝜃c with both reflected and transmitted light, and the yellow vectors represent the case of 𝜃i > 𝜃c and total internal reflection.

It should be noted that air-incident light with a grazing incident angle of 𝜃i = 90deg is transmitted into the water at the critial angle: 𝜃t = 𝜃c. Thus light from the entire sky is transmitted through the surface into a cone of half angle 𝜃c, which is known as Snel’s cone.

Fresnel’s Equations for Unpolarized Light

The equations of the previous section show the relations between the angles and directions of the incident and the reflected and transmitted light. However, they do not show how much light is reflected or transmitted. That information is given by Fresnel’s equations.

Consider a collimated beam of unpolarized incident light, which has some irradiance measured on a surface normal to the direction ξ̂ i of propagation. The fraction of this incident irradiance that is reflected by the air-water surface is

RF (𝜃i) = 1 2 sin(𝜃i 𝜃t) sin(𝜃i + 𝜃t)2 + tan(𝜃i 𝜃t) tan(𝜃i + 𝜃t)2 , (5)

which holds for 𝜃i0. For normally incident light, 𝜃i = 0, the reflectance is

RF (𝜃i = 0) = nw 1 nw + 12. (6)

Equations (5) and (6) hold for both air- and water-incident light. Given the incident angle 𝜃i, the transmitted angle 𝜃t is computed using either Eq. (2) or (3), and then Eq. (5) (or 6) can be evaluated. For water-incident light and 𝜃i 𝜃c, RF = 1. Figure 3 shows the Fresnel reflectance for the range of water indices of refraction at visible wavelengths.

To be completely general, the Fresnel equations should use the complex index of refraction m = n + ik, where n is the real index of refraction seen above and k(λ) = λa(λ)2π is the complex part (a is the absorption coefficient). Thus Eq. (6) should be

RF (𝜃i = 0) = m 1 m + 12 = (nw 1)2 + k2 (nw + 1)2 + k2.

However, for water at near-UV to near-IR wavelengths, k < 106 and the difference is negligible. However k can be of order 0.1 to 1 at some UV and far-IR wavelengths, in which case the complex index of refraction must be used.


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Figure 3: Fresnel reflectance for unpolarized light and selected water indices of refraction.

Conservation of energy requires that the sum of the reflected and transmitted energy equal the incident energy. Thus the fraction of the incident energy that is transmitted is TF = 1 RF . It can be confusing to see that energy is conserved when different quantities such as plane irradiance, scalar irradiance, or radiance are used to describe the light, or when the incident light is not a single collimated beam. This is discussed in detail on the page on energy conservation.

Reflection and transmission are much more complicated when the incident light is polarized. The Fresnel reflectance and transmittance equations for polarized light are given in the Level 2 material for this chapter.

The n2 Law for Radiance

Snel’s law yields an important result governing how unpolarized radiance changes when going from one medium to another, e.g., when crossing an air-water surface. Figure 4 shows two beams of radiance, one incident onto an interface and one transmitted. Let L1 be the incident radiance in medium 1 defined by power Φ1 passing through an area ΔA1 normal to the direction of photon travel and contained in a solid angle ΔΩ1 = sin𝜃1Δ𝜃1Δϕ1, where 𝜃1 is polar angle measured relative to the normal to the surface and Δϕ1 is the width of the solid angle in the azimuthal direction. Likewise, L2 is the transmitted radiance in medium 2 defined by the corresponding quantities as illustrated. The azimuthal angle does not change when crossing the surface, so ΔΩ2 = sin𝜃2Δ𝜃2Δϕ1. The incident and transmitted power passes through a common area ΔA at the interface.


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Figure 4: Geometry for deriving the n2 law for radiance.

The indices of refraction n1 and n2 are fixed, but the polar angle 𝜃 changes when crossing the interface. Squaring Eq. (1) and differentiating gives

n12 sin𝜃 1 cos𝜃1Δ𝜃1 = n22 sin𝜃 2 cos𝜃2Δ𝜃2.

Multiplying each side of this equation by the common value of Δϕ and rewriting in terms of solid angles gives

n12 cos𝜃 1ΔΩ1 = n22 cos𝜃 2ΔΩ2,

which is known as Straubel’s invariant.

The radiances are defined by

L1 = ΔΦ1 ΔA1ΔΩ1andL2 = ΔΦ2 ΔA2ΔΩ2.

Fresnel’s equation gives the transmitted power as ΔΦ2 = [1 RF (𝜃1)]ΔΦ1 = TF ΔΦ1. The areas are related by ΔA1 = ΔAcos𝜃1 and ΔA2 = ΔAcos𝜃2. Thus the ratios of the incident and transmitted radiances can be written as

L2 L1 = ΔΦ2 ΔΦ1 ΔA1ΔΩ1 ΔA2ΔΩ2 = TF cos𝜃1ΔΩ1 cos𝜃2ΔΩ2 = TF n22 n12

or

L2 n22 = TF L1 n12.

This result is called the n-squared law for radiance. The quantity Ln2 is sometimes called the reduced radiance or the basic radiance.

Although energy is conserved when crossing a boundary, the radiance changes by a factor proportional to the change in the index of refraction squared. This is a simple consequence of the change in solid angle resulting from the change in 𝜃 when crossing the boundary. Note that for normal incidence and nw = 1.34, TF 0.979 and the radiance just below a water surface is 0.979(1.34)2 1.76 times the radiance in the air. Conversely, when going from water to air, the in-water radiance is reduced by a factor of 1.76.

To the extent that losses to absorption and scattering out of the beam can be ignored (sometimes a good approximation for atmospheric transmission, but almost never the case in water), the radiance divided by the square of the index of refraction is constant along any path. This result has even been called the fundamental theorem of radiometry, which is perhaps a bit grandiose given that real beams always lose radiance due to absorption and can lose or gain radiance due to scattering.

Finally, note that the n2 law applies only to radiance transmission. When tracing photons in a Monte Carlo simulation, from which the radiance can be estimated by appropriate binning of the transmitted photons, no n2 factor is applied to the energy of the transmitted photons or to the radiance estimated from the detected photons. This is because the n2 effect is automatically built into the radiance estimate photon by photon as the directions of the individual photons are computed by Snel’s law.

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