The wavelength of light in the visible range is normally very small (on the order of 10−7 meters) in relation to the size of the optical features of interest, which include surface boundaries, volumes of space, and lenses. In these cases, to a first approximation, it is possible to model the behavior of wavefronts by neglecting wavelengths. Thus, in the limit thatλgoes to zero, light may be modeled in terms of geometrical behavior. When this is done, we speak of geometrical optics.
In this section, we assume that regions are free of currents and charges, i.e.
j= 0 andρ= 0. The general form of the fields considered here is
E= e(r)eiβ S(r), (2.172a)
H= h(r)eiβ S(r), (2.172b)
where S(r) is a real scalar function of space r. The function S(r) is known as the optical path or the eikonal. Vectors e(r) and h(r) are complex vector valued functions of space. Maxwell’s equations for this particular solution can then be
2.9. Geometrical Optics 85
rewritten using the identities given in Section A.9 as (∇S)×h +εe= −1 In the limit that the wavelength goes to zero, the value ofβ goes to infinity (be-causeβ =2λπ). As a result Maxwell’s equations simplify to
(∇S)×h +εe= 0, (2.174a)
(∇S)×e −μh= 0, (2.174b)
e· (∇S) = 0, (2.174c)
h· (∇S) = 0. (2.174d)
If h from (2.174b) is substituted into (2.174a), the latter equation yields 1
μ
(e · (∇S))∇S − e(∇S)2
+εe= 0. (2.175)
Substituting (2.174c) into this equation gives
(∇S)2=μ ε, (2.176a)
This equation is known as the eikonal equation. The gradient of S can be seen as the normal vector of the wavefront. It can be shown that the average Poynting vector S is in the direction of the wavefront:
S=∇S
n . (2.178)
2.9.1 Fermat’s Principle
The eikonal equation provides a description of geometrical optics. An alternative description is afforded by Fermat’s principle, which states that light follows a trajectory (a ray) such that the optical path length is an extremum. The optical path length is defined as
b a
n ds, (2.179)
where ds is an element of arc length, and n is the index of refraction. The path has fixed end points a and b. Minimizing this integral can be achieved using the calculus of variations [736, 1082] resulting in
d
This equation is known as the ray equation [466, 617, 650, 1081] and is valid for inhomogeneous isotropic media that are stationary over time. This equation expresses the fact that at every point in the medium, the tangent and the normal vector associated with the ray path span a plane called the osculating plane, and the gradient of the refractive index must lie in this plane.
A consequence of Fermat’s principle is that in a homogeneous medium where n(r) is constant, light travels in a straight line. In inhomogeneous media, however, light may travel along curved arcs according to the above ray equation. This happens, for instance, in the atmosphere, as shown in Section 2.10.4. Finally, Snell’s law for reflection and refraction can be derived from Fermat’s principle (Section 2.5.1).
2.9.2 The Cosine Law of Illumination
The cosine law of illumination states that a surface can receive radiation only in proportion to its area projected in the direction of the light source. If the direction of the light source is given by L and the surface normal is N, then the cosine of the angle of incidenceθis given by their dot product:
cos(θ) = N · L. (2.181)
The projected area is then
A= A N · L = A cos(θ), (2.182) as illustrated in Figure 2.52. In this figure, a large surface is angled away from the light source, whereas a smaller surface is aimed directly at the light source. The
2.9. Geometrical Optics 87
θ N
A A’
A’ = A cos(θ)
Figure 2.52.A surface can only receive radiation proportional to its projected area in the direction of the light source.
projected area of the larger surface in the direction of the light source is identical to the projected area of the smaller surface, and therefore both surfaces receive the same amount of radiation.
A similar law called the cosine law of emission states that radiation emit-ted from iso-radiant or Lambertian surfaces8decreases with a factor of cos(θ), whereθis the angle of exitance. An example is shown in Figure 2.53, where the radiation emitted from a point decreases as the angle of exitance increases.
N θ1
θ2
Figure 2.53. The radiation emitted from a uniform diffuser decreases as the angle of exitanceΘ increases.
Surfaces that obey the cosine law of emission appear equally light regardless of the viewing direction. Although no perfect Lambertian surfaces exist in prac-tice, many surfaces are well approximated by this law. These are called matte or diffuse surfaces.
2.9.3 The Inverse Square Law
In Section 2.1.1, we saw that according to Coulomb’s law two charges exert a force on each other that diminishes with the square of their distance. The cause
8Strictly speaking, these surfaces are iso-radiant. Although a surface might be Lambertian at each point individually, there may be spatial variation in the amount that is emitted. Iso-radiance requires uniformity across the entire surface [839].
Point source
Wavefronts
Figure 2.54.Wavefronts diverging spherically from a point source.
of this relation lies in a more general law which stems from geometrical consid-erations; this more general law is called the inverse square law9.
Consider a point source that emits light. This light propagates through space as spherical wavefronts. The farther the wave has traveled, the greater the surface area spanned by the wave. The energy carried by the wave is thus distributed over increasingly large areas, and therefore the energy density is reduced (Figure 2.54).
Figure 2.55.The objects farther from the light source (a 25W omni-directional incandes-cent light) are dimmer than the objects closer to it due to the inverse square law.
9This law applies to other modalities as well, including gravity and sound.
2.9. Geometrical Optics 89
The inverse square law states that the surface density of radiant energy emitted by a point source decreases with the squared distance d between the surface and the source:
Ee∝ 1
d2. (2.183)
An example is shown in Figure 2.55 where the mugs farther from the light source are dimmer than the objects nearer to it.
As a general rule of thumb, if the distance of an observer to a source is greater than 10 times the largest dimension of the source, it can be approximated as a point source with respect to the observer [397]. The error associated with this approximation is important in radiometry and will be further discussed in Sec-tion 6.2.11.
2.9.4 Bouguer, Beer, and Lambert’s Laws
Light traveling through a dielectric medium will occasionally collide with a par-ticle and either undergo scattering, or be absorbed and turned into a small amount of heat. Both phenomena are discussed in detail in the following chapter. How-ever, the effect of either result of the collision is that a photon has a probability of interacting with the medium for every unit of distance traveled.
Hence, the irradiance after traveling a distanceΔs through the dielectric ma-terial will be attenuated by some amount:
Ee(s + Δs) = Ee(s) (1 −σaΔs). (2.184) Here,σais a wavelength-dependent quantity modeling the attenuation induced by the medium. It is known as the spectral absorbtivity. Rewriting this equation and taking the limitΔs → 0, we find Assuming that the medium is homogeneous and isotropic, this equation can be solved to yield
Ee(s) = Ee(0) e−σas. (2.186) Thus, the attenuation in an absorbing dielectric material is exponential. This result is known as Bouguer’s law. The ratio Ee(s)/Ee(0) is called the transmittance T:
T= Ee(s)
Ee(0) (2.187a)
= e−σas. (2.187b)
A related quantity is absorbance A, written as
A= −lnEe(s)
Ee(0) (2.188a)
=σas. (2.188b)
In spectroscopy and spectrophotometry, transmittance and absorbance are often expressed in powers of 10. The conversion between the two forms is a factor of ln(10) ≈ 2.3:
T= 10−σas/ln(10), (2.189a)
A=σas/ln(10). (2.189b)
The attenuationσacan be split into a material- dependent constantε, which is called either molar absorbtivity, extinction coefficient, or the concentration of the absorbing material c. The absorbance is then given as
A=εcs, (2.190)
and the transmittance is
T= 10−εcs. (2.191)
This relation is known as Beer’s law. Here, we have merged the ln(10) factor with the constant ε. As both concentration c and pathlength s have the same effect on absorbance and transmittance, it is clear that a given percent change in absorbance or transmittance can be effected by a change in pathlength, or a change in concentration, or a combination of both.
Beer’s law is valid for filters and other materials where the concentration c is low to moderate. For higher concentrations, the attenuation will deviate from that predicted by Beer’s law. In addition, if scattering particles are present in the medium, Beer’s law is not appropriate. This is further discussed in Section 3.4.6.
Finally, if the pathlength is changed by a factor of k1from s to k1s, then the transmittance changes from T to Tas follows:
T= Tk1, (2.192)
known as Lambert’s law. Similarly, if the concentration is changed from c to k2c, the transmittance goes from T to T:
T= Tk2. (2.193)
2.9. Geometrical Optics 91
Specular surface Rough surface
Surface normals Light paths
Figure 2.56.A perfect specular reflector (left) and a rough surface (right). Light incident from a single direction is reflected into a single direction for specular surfaces, whereas rough surfaces reflect light into different directions.
2.9.5 Surface Reflectance
We have shown that when a plane wave is incident upon a boundary between two dielectrics, two new waves are created. These new waves travel in the reflected direction and the transmitted direction. For a boundary between a dielectric and a conductive object, such as a metal, the refracted wave is dampened and, there-fore, travels into the material for a short distance. Reflected and refracted waves traveling in directions as predicted by Snell’s law are called specular.
There are several aspects of material appearance that are not explained by this model. For instance, it is not clear why shining a white light at a tomato produces a red color; nor are diffuse, glossy, or translucent materials explained. While the color of an object depends on the atomic structure of the material (this is the topic of Chapter 3), non-specular surface reflectance and transmittance are the cause of microscopic variations in surface orientation.
Thus, the occurrence of different levels of shininess depends on surface fea-tures that are larger than single atoms and molecules, but still below the resolving power of the human visual system. Specular reflection can, therefore, only occur when the boundary is perfectly smooth at the microscopic level, i.e., the surface normal is constant over small surface areas. At the macroscopic level, the surface normal may still vary; this is then interpreted by the human visual system as a curving but smooth surface.
Multiple reflections
Surface normals Light paths
Figure 2.57.Self-occlusion occurs when reflected rays are incident upon the same surface and cause a secondary reflection.
A rough surface has surface normals pointing in different directions. Since at each point of the surface, light is still reflected specularly, shown in Figure 2.56, a beam of light is reflected into many different directions. The appearance of such an object is glossy. In the limit that the surface normals have a uniform distribution over all possible directions facing away from the object, light will be reflected in all directions equally. Such surfaces are perfectly diffuse and are called Lambertian surfaces.
The distribution of surface undulations can be such that reflected rays are in-cident upon the same surface before being reflected again away from the surface.
Such self-occlusion is shown in Figure 2.57.
2.9.6 Micro-Facets
The most direct way to model the macroscopic reflectance behavior of surfaces is to treat a surface as a large collection of very small flat surfaces, called micro-facets. It is generally assumed that each facet is perfectly specular, as outlined in the preceding section. The orientation of each facet could be explicitly modeled, but it is more efficient to use a statistical model to describe the distribution of orientations. An example is the first micro-facet model, now named the Torrance-Sparrow model [1140].
Blinn proposed to model the distribution of orientations D(Θi,Θo) as an expo-nential function of the cosine between the half-angle and the aggregated surface normal [105]. The half-angle vectorΘhis taken between the angle of incidence Θiand the angle of reflectionΘo:
D(Θi,Θo) = (Θh· n)e (2.194a) Θh= Θi+ Θo
Θi+ Θo . (2.194b)
The exponent e is a user parameter.
Different micro-facet models make different assumptions, for instance, on the existence of self-occlusion. Another common assumption is that micro-facets form a regular grid of V-shaped grooves. One model, namely Oren-Nayar’s, as-sumes that each facet is a perfectly diffuse reflector, rather than a specular reflec-tor [859]. In this model, the amount of light reflected off a surface in a particular directionΘoas function of light incident from directionΘiis given by
fr(Θi,Θo) =ρ
π(A + B max(0,cos(φi−φo) sin(α) tan(β))), (2.195a) A= 1 − σ2
2σ2+ 0.66, (2.195b)
2.9. Geometrical Optics 93
B= 0.45σ2
σ2+ 0.09, (2.195c)
α= max(θi,θo), (2.195d)
β= min(θi,θo). (2.195e)
In this set of equations,σ is a free parameter modeling the roughness of the sur-face. The factorρ is the (diffuse) reflectance factor of each facet. The directions of incidence and reflection (ΘiandΘo) are decomposed into polar coordinates, where polar angleθ is the elevation above the horizon and azimuth angleφis in the plane of the facet:
Θ = (θ,φ). (2.196)
The above micro-facet models are isotropic, i.e., they are radially symmetric.
This is explicitly visible in the Oren-Nayar model where the result depends only on the difference between φi andφo, but not on the actual values of these two angles. This precludes modeling of anisotropic materials, where the amount re-flected depends on the angles themselves. A characteristic feature of anisotropic materials is, therefore, that rotation around the surface normal may alter the amount of light reflected. An often-quoted example of an anisotropic material is brushed aluminium, although it is also seen at a much larger scale for exam-ple in pressed hay as shown in Figure 2.58. Several micro-facet models that take anisotropy into account are available [45, 46, 702, 921, 1005, 1212].
Figure 2.58. Pressing hay into rolls, as shown here, produces an anisotropic effect on the right side due to the circular orientation of each of the grass leaves; Rennes, France, June 2005.
2.9.7 Bi-Directional Reflectance Distribution Functions
A further abstraction in the modeling of surface reflectance is afforded by bi-directional reflectance distribution functions, or BRDFs for short [838]. The micro-facet models presented in the preceding section can be seen as a specific class of BRDF. In general, BRDFs are functions that return for each possible an-gle of incidence and anan-gle of reflectance, the fraction of light that is reflected at the surface.
For BRDFs to be plausible models of physical reality, at the very least they should be reciprocal as well as energy conserving. Reciprocal means that if the angle of incidence and reflectance are reversed, the function returns the same value:
fr(Θi,Θo) = fr(Θo,Θi). (2.197) The energy-conserving requirement refers to the fact that light incident upon a sur-face is either absorbed, transmitted, or reflected. The sum of these three compo-nents can not be larger than the incident amount. Light emitted by self-luminous surfaces is modeled separately and does not form part of a BRDF.
For computational convenience, the reflection of light may be classified into four broad categories: diffuse, glossy specular, perfect specular, and retro-reflective, although most surfaces exhibit a combination of all four types [895].
Fully specular BRDFs behave like the reflection detailed in Section 2.5, i.e., the Poynting vector of incident light is mirrored in the surface normal to determine the outgoing direction [666]:
fr(Θi,Θo) =δ(θo−θi)δ(φo−φr−π)
sin(θr) cos(θr) , (2.198) where we have used (2.196), andδ() is the Kronecker delta function. At the other extreme, Lambertian surface models assume that light incident from all different directions is reflected equally [645]:
fr(Θi,Θo) = 1
π. (2.199)
Although this is not a physically plausible model, it is a reasonable approximation for several materials, including matte paint.
An early model of glossy reflection was presented by Phong [897] and is given here in modified form with the coordinate system chosen such that x and y are in the plane of the surface and the z-coordinate is along the surface normal [895]:
fr(Θi,Θo) =
Θi· (−Θox,−Θoy,Θoz)T e
. (2.200)
2.9. Geometrical Optics 95
Here, the outgoing directionΘo is scaled first by(−1,−1,1)T, thus mirroring this vector in the surface normal. It is now possible to make this scaling a user parameter, allowing the modeling of various types of off-specular reflection:
fr(Θi,Θo,s) = Θi·
sΘTo
e
. (2.201)
While this BRDF could be used to model glossy materials, there is a further re-finement that can be made by realizing that this function specifies a single lobe around the angle of incidence. More complicated BRDFs which are still physi-cally plausible can be created by modeling multiple lobes and summing them:
fr(Θi,Θo) =ρd
π models a diffuse component. Each of the n lobes is modeled by a different scaling vector si as well as a different exponent ei. The result is called the Lafortune model [637] .
An empirical anisotropic reflectance model takes into account the orientation of the surface. The amount of light reflected intoΘodepends on the angle of rota-tionφnaround the surface normal. Defining directional vectors for the incoming and outgoing directions (vΘi and vΘo), the half-vector vhis defined by
vh= vΘi+ vΘo
vΘi+ vΘo . (2.203)
The angleΘnhis between the half-vector and the surface normal. Ward’s aniso-tropic BRDF is then given by [269, 1200, 1212]
fr(Θi,Θo,s) =ρd
The specular and diffuse components are modeled byρsandρd. Their sum will be less than 1. The function is parameterized inσx andσy, each of which are typically less than 0.2. These two parameters quantify the level of anisotropy in the model. If they are given identical values, i.e.,σ =σx=σy, the model simplifies to an isotropic model:
fr(Θi,Θo,s) = ρd
Material ρd ρs σx σy
Lightly brushed aluminium 0.15 0.19 0.088 0.13
Rolled aluminium 0.10 0.21 0.04 0.09
Rolled brass 0.10 0.33 0.05 0.16
Enamel finished metal 0.25 0.047 0.080 0.096 Table 2.5.Example parameter settings for Ward’s model of anisotropic reflection (After [1212]).
Parameters measured for several materials are listed in Table 2.5.
Although BRDFs are typically modeled as the ratio of incident to reflected light and depend on incoming and outgoing directions, they can be extended to be wavelength dependent, i.e., fr(Θi,Θo,λ). In addition, polarization can be accounted for in formulations related to BRDFs; these formulations are then called bi-directional surface scattering reflectance distribution functions (BSS-RDF) [36].