A certain class of microstructured surfaces in 2D geometry is studied on the basis of an approximation of infinitesimal microstructure elements. The macroprofile of the surfaces is then treated as a new type of optical surface with a deflection law that differs from the law of reflection and Snell’s law. We discuss the propagation of wavefronts by general microstructured surfaces and the discontinuity of the eikonal function at the microstructure. Naturally, a classification of the microstructures is obtained (regular and anomalous), and the concept of 2D ideal microstructures is also introduced, since they are used for coupling two macroscopic extended bundles.
©2006 Optical Society of America
Structured optical surfaces, as Fresnel lenses, TIR lenses or Fresnel mirrors, are often used in nonimaging optical design . When the structure element is small enough (microstructured surface), it can be approximated as infinitesimal for some calculations, and the macro-profile of the surface can be treated as a new type of optical surface with a certain deflection law, which will be different of the reflection law or the Snell law. This asymptotic limit has been widely used previously for designing Fresnel lenses in single surface designs [1–4], and also for TIR lenses in combination with other surface [5–9]. In this paper we will consider a class of generalised microstructures in that limit, and we will study its properties.
As an first example, consider the Fresnel-like mirror shown in Fig. 1(a), in which the slope of these mirrors relative to the tangent vector is given by the angle α, which in general is a function of the parameter s along the line (α=α(s)). If the facet size is consider to be infinitesimal, the resulting surface can be represented as shown in Fig. 1(b), i.e., as a line with reflects the rays with a non-conventional reflection law given by θ d=θ i-2α.
In general, we define a microstructured optical surface in two dimensions as a line where the rays suffer a deflection, where deflection here means a change of the ray vector direction. A class of microstructured surfaces can be characterized by its “law of deflection” which gives the direction of the ray after its deflection as a function of the direction of the incident ray and the unit normal to the line. For instance, if P i is the unitary incident ray vector, P d is the unitary deflected ray vector and N is the unitary normal to the line, then a law of deflection is a function P d=F(P i, N, ϕ), where ϕ is n-dimensional vector that constitutes a parameter that is defined by the specific microstructure. The microstructured surface will be transmitive or reflective type depending on incident and exiting rays laying at the same or opposite side of the microstructured line, respectively (so the example in Fig. 1 is reflective).
We will restrict this study to the case in which function F is continuous and is such that the exit angle θ d varies monotonically with the incident angle θ d , so the sign of dθ d /dθ i does not change. This restriction excludes of this analysis several types of microstructures that have been used previously for a variety of purposes, such as providing gaps in CPC collector designs by controlled directional scattering ], improving the performance of rotational symmetric optics with non-rotational microstructures [11–12], or light homogenization . The generalization of the use of the approach presented here to cover those microstructures would be interesting due to their relevance, but it is beyond the scope of this paper.
Because the function F may not be defined for all possible values of P i, N, ϕ, the set of permitted values is called the domain of definition of the deflection law. In Fig. 1 the parameter ϕ=α is one-dimensional and the deflection law is P d=P i-2(N m·P i)N m, where N m is the normal to the small mirror facet, which is a function N m=N m(N,α).
Therefore, a microstructured surface can be characterized by: (i) its law of deflection, (ii) the equations of the line in parametric form x=x(s), z=z(s), (s is a parameter along the line) and (iii) the function ϕ=ϕ(s). Contrary to aforementioned law relating to microstructured surfaces, the law of deflection in the case of non-microstructured surfaces does not depend on a parameter ϕ, that is, P d=F(P i, N). Therefore, in general, the microstructured surfaces provide another degree of freedom. The equations x=x(s), z=z(s) will be called the microstructured surface equation.
2. Discontinuity of the eikonal function at the microstructured surface
Fig. 2 shows a region of the x-y plane, in which there are known trajectories of two one-parameter bundle of rays, one of them (B 1) impinging on, and the other one (B 2) exiting, a given microstructured optical surface (i.e., the deflection law is known). Assume also that at any point of this region there is one and only one ray per bundle crossing the optical surface.
We can construct the wavefronts of each bundle (these are just the lines normal to the rays) and thus we can calculate the eikonal functions O 1(x, y) and O 2(x, y) which give, respectively, the optical path length from the point (x, y) to a given reference wavefront for the bundle B 1 and B 2. Consider also the trajectory of a single ray of B 1 that is deflected at the point P and, after the reflection, becomes a ray of B 2.
Consider that the wavefronts used as reference for calculating O 1 and O 2 pass through P. This selection leads to the simplest expressions below. The use of other arbitrary reference wavefronts would just add a trivial constant to some of the expression below).
In the case of non-microstructured surfaces, the equation of the line is simply given by stating the continuity of the eikonal function, i.e. by the equation O 1(x, y)=O 2(x, y), which is equivalent to the Fermat Principle.
Let us define the following function:
and consider the parameterization of the line such point P corresponds to s=0. Then, by the selection of the reference wavefronts D(0)=0.
When the line is not microstructured, then O 1(x, y)=O 2(x, y) along all the line and then D(s)=0. However, in the microstructured case, the additional degree of freedoms in the deflection law is going to lead to a more general result. Deriving Eq. (1):
where the ∇ symbol denotes gradient operator, the dot on the variables denotes derivation with respect to s, n i and n d are the refractive indices of the medium where B 1 and B 2 proceed, respectively. θ i =cos-1(-P i ·N). On the other hand, θ d =cos-1(-P d·N) for transmitive lines while θ d =cos-1(P d ·N) for the reflective ones. Eq. (2) can be integrated as:
which indicates that the eikonal function O 1(x, y)) and O 2(x, y) are not the same at the line, i.e., in general, there is a discontinuity in optical path length at a microstructured optical surface. In the non-microstructured case, in which D(s)=0, Eq. (2) vanishes, which just leads to the Snell law (and the reflection law when n i =n d ).
3. Etendue considerations and the edge-ray theorem
In general, not all the rays impinging on a real microstructured surface are to be deflected according to the law. This can be due to the actual design of the microstructure or, when dealing with extended ray bundles, due to more fundamental reasons derived for the etendue conservation theorem, as was discussed in Ref.  and Ref. .
Consider the specific case of a microstructure for transmitting an extended ray bundle impinging on a portion of line ds forming angles with respect to the normal vector to the line within θ i (s) and (s) and exiting it within θ d (s) and (s) (see Fig. 3). Without loss of generality, we can assume consider θ i (s) > (s) (on the contrary, just rename the angles). By application of the deflection law (which we stated to be monotonous), the deflected rays will form angles within θ d (s) and (s). The differential etendue of the impinging bundle can be written as dE i =n i ds(sinθ i -sin ) and, in general, it does not coincide in general with the etendue of the deflected bundle dE d =n d ds|sinθ d -sin |. The absolute value is needed because the deflection law F being monotonous implies that the sign of dθ d /dθ i does not change, so dθ d /dθ i may be non-negative or non-positive (and in this last case θ d (s)< (s).
We will call lossless microstructures to those in which all the impinging rays of the bundle are deflected according to the law. For theses microstructures, it is fulfilled that dE i ≤dE d . Note that the converse is not true, i.e., a microstructure can fulfill dE i ≤dE d and not be lossless as defined. Note also that a microstructure can be lossless for one specific incident extended bundle but to have losses for other different incident bundles.
Analogously, we will call fully filled microstructures to those in which all the deflected rays come from incident rays according to the law. For theses microstructures, it is fulfilled that dE i ≥dE d . Note that if we reverse the ray directions, fully filled microstructred are lossless, and vice versa. We will call ideal microstructures to those which are losses and fully filled. The ideal microstructures fulfill the equality dE i =dE d . Microstructures are not lossless because they have leakage , shading or blockage [14, 4] of rays, and microstrustructures that are not fully filled are said to present output gaps  or dilution .
Let us classify the microstructures into two groups, the regular type for which θ d (s)> (s) and the anomalous type for which θ d (s)< (s). Being regular or anomalous will depend on the specific microstructure and its corresponding deflection law. Thus regular deflection laws have dθ d /dθ i ≥0 while anomalous ones have dθ d /dθ i ≤0. The same classification applies for reflecting or transmitting microstructures (see Fig. 3.). According to this classification, non-microstructured surfaces (i.e., conventional refractive surfaces and mirrors) are of the regular type.
For regular ideal microstructured surface, assuming that the four reference wavefronts involved pass through the same point P of the line (so D(0)=D’(0)=0), we deduce:
This is the necessary condition (but not sufficient) for a regular microstructure to be ideal for that extended bundle is that the discontinuity along the microstructure surface of the eikonal functions associated to the two edge ray bundles coincide. In the case of non-microstructures surfaces, this is automatically fulfilled since both D(s) and D ’(s) are null.
As an application example of this concept let us calculate the (necessary) profile that an ideal regular microstructured surface would have to deflects an incoming extended bundle limited by the directions v=(cos α, sin α) and v ’=(cos α,-sin α) into the bundle crossing the segment AA’ (see Fig. 4). It is simple to see that at the point X=(x, y), O 1(X)=X·v, O 2(X)=|X-A|, (X)=X·v and (X)=|X-A ’|. Then D(X)=X·v-|X-A| and D ’(X)=X·v’-|X-A’|. From Eq. (4), D=D’ leads to:
It is easy to obtain that (X-A ′-X-A)(X-A′+X-A)=2A-A ′ y. Then, Eq. (5) can be written as |X-A ′|+|X-A|=|A-A ′|/sinα, which states that the line is an ellipse with foci A and A′. This result is consistent with the already known result that a curved thin nonimaging Fresnel lens, at the limit in which its refractive index is infinite, becomes ideal and follows the ellipse [15–16].
On the other hand, in the case of anomalous microstructures, we get:
where S(s) is the function S(s)=O 1(x(s),y(s))+O 2(x(s),y(s)).
It is immediate to check that the calculation of the profile of an anomalous ideal microstructured surface for the previous example leads to the same ellipse of Fig. 4. However, the study of rotational symmetric microstructured surfaces  reveals the very different properties of regular and anomalous microstructures in 3D.
Finally, consider the problem of coupling two extended ray bundles Mi and Mo (for short, Mi=Mo). The edge ray theorem, which applies for non-microstructured surfaces, states that only the edge rays ∂Mi and ∂Mo need to be coupled (∂Mi=∂Mo). According to the previous definitions, if an optical system with ideal microstructured surfaces is designed to couple the edge rays ∂Mi and ∂Mo also Mi=Mo. If the microstructure is not ideal but lossless, then the edge ray coupling will lead to Mi⊂Mo and if it is fully filled, Mi⊂Mo. In this sense, the edge ray theorem has been applied to the nonimaging design with microstructured surfaces [6, 8].
4. Examples of real microstructures
The parameter function ϕ(s) only partially defines the “real” microstructure. For instance, for the microstructure surface shown in Fig. 1, ϕ≡α is one-dimensional and thus only the active facet (from the design point of view) is defined. To be complete, the definition of these microstructures also needs to describe the tilting angle of the inactive facet, which physically joins adjacent active facets.
A thin Frensel lens is an example of regular microstructure. If the slope of the two refractive facets are to be designed [1, 3 18], the parameter ϕ=(α, α ’), is two-dimensional, where angles α and α ’ are the angles formed by the refractive facets with the tangent to the microstructure line, respectively. These parameters has been fixed in the past either to couple the edge rays of a extended bundle [1, 18], or to produce minimum chromatic aberration [3, 18]. As in the case of the Fresnel mirror (or a single sided Fresnel lens), there is also an inactive facet that must be selected to complete the definition of the real microstructure. Usually this facet is design to make the microstructure lossless, although in some designs this is not compatible with the additional restriction of making it demoulding in an injection molding process is set.
Ideal Fresnel lens microstructures are obtained when one of the two angles α or α ’ is null (i.e., one facet is parallel to the tangent vector to the microstructured line) and the other angle is selected to make its corresponding refraction normal to the flow line of the extended bundle [see Fig. 5.(a)–5(b)]. The flow line is the line bisecting the extended bundle . The “inactive” facet is set as a mirror parallel to the flow line, which has the property of guiding the bundle without disturbing it. (Note that this flow line mirror will reflect some of the rays, which strictly doesn’t match with the definition of the deflection law as stated above). The “inactive” facet performs as a perfect mirror for incidences from the interior of the material (by TIR) and as a food mirror for the exterior incidences (because the Fresnel reflection at grazing incidence is high). In any other of the aforementioned two ideal cases, the Fresnel microstructure cannot be ideal even in the asymptotic case in which the angular spread of the transmitted bundle tends to zero. Note that in the ideal Fresnel lens microstructure a single refraction deflects the flow lines, so its capability for deflecting bundles is similar to that of a single refraction.
An example of anomalous microstructure is a TIR lens. In the case of general 3 TIR facet designs, the parameter ϕ is three-dimensional. However, if we restrict the analysis to TIR lenses with only two facets (for instance, with different refractive indices at the input and the output, see Fig. 5.(c)–5(d), then ϕ is only two-dimensional: ϕ=(α,α ’), where angles α and α ’ are the angles formed by the facets with the tangent to the microstructure line, respectively. Usually the facet of angle α ’ is fixed by the condition of performing as a TIR flow line mirror for the extended bundle, or by the condition of being demoldeable (usually not compatible with the flow line condition). In some applications is also interesting to make the microstructure to be lossless and as much filled as possible. This second condition fixes the angle α as indicated in Fig. 5.(c)–5(d) differently for the TIR lens working for a receiver and for an emitter . The flat facet TIR lens microstructures tend to be ideal at that asymptotic limit in which the angular spread of the transmitted bundle tends to zero. This makes TIR facets to have theoretically a higher potential to get closer to the thermodynamic limits when needed.
The analysis of microstructured surfaces that produce a continuous deflection of the rays incident on it has been presented. From the theoretical point of view, the propagation of wavefronts through them, with the corresponding calculation of the discontinuity of the eikonal at the microstructured line, and the study of propagation of extended bundles, with the appearance of a natural classification of the microstructures (regular and anomalous) and the introduction of the concept of ideal microstructure.
The profile of the ideal microstructured line (either regular or anomalous) is proven to be an ellipse only by etendue considerations, and real regular and anomalous microstructures (an ideal Fresnel lens facet and a TIR lens facet, respectively) are presented.
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