1. Introduction

The cylindrical V-groove reflector is a well known optical device, and its use is spread to several applications, particularly signaling and displays. Figure 1(a) shows its cross section, formed by two flat profiles that orthogonally join at the groove peak. One important property of this reflector configuration is that an incident ray in two dimensional geometry (2D), with direction cosines (p,q), that reflects at both sides of the reflector, ends up with direction cosines (-p,-q), thereby reversing the ray’s trajectory. Thus the linear reflector is a perfect retroreflector for 2D planar wavefronts.

 

Fig. 1 Examples of groove reflectors (a) The flat linear 90° reflector is a perfect retroreflector in 2D of plane wavefronts (b) The carambola reflector is a perfect retroreflector for a spherical wavefront.

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Another example of retroreflector is that of Fig. 1(b), which was described in reference [1] for brightness enhancement of light sources. The device is a combination of n equal pieces equiangularly disposed around the center A [in Fig. 1(b), n = 5]. Each piece comprises two symmetric confocal parabolas, with their common focus at point A. The symmetric confocal parabolas form a perfect retroreflector for a single point source. This retroreflector configuration was also proposed in reference [2].

In this paper we state a more general design problem for two arbitrary given wavefronts, and find that there exist solutions with analytic profiles (i.e. admit Taylor series expansion). These solutions [3] are calculated by the Simultaneous Multiple Design method in two dimensions (SMS2D).

Another example of a V-groove reflector is the parabolic retroreflector known as the

2. Statement of the problem

There are two possible design problems, as defined in Fig. 2 . The first design, herein called reflector Type I, represents a retroreflector for two wavefronts. Each wavefront is converted into itself after two reflections at both sides of the groove. The second design, herein called reflector Type II, provides perfect coupling between the rays of two wavefronts. In both cases, two profiles at adjacent sides of the groove peak need to be designed. In particular cases, both profiles can be symmetric with respect to a line passing through the groove peak (the axis y) but, in general, the profiles are not symmetric.

 

Fig. 2 Definition of design problems (a) Type I (b) Type II.

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These design problems can be formulated in the framework of Functional Differential Equations, as discussed in reference [4]. For this formulation we will consider first the particular case of two spherical wavefronts for the Type I reflector. The implementation of an analogous formulation for a Type II reflector is straightforward.

Figure 3 presents an asymmetric Type I reflector. Without loss of generality, we can select the coordinate system such that the groove peak has coordinates (0, h) and that two centers of the wavefronts are points (x_A, 0) and (-x_B, 0), where x_A, x_B and h are input parameters of the design. Let us describe the reflector profiles by two functions f(y) and g(y). A ray leaving source A reflects on the first profile at the point (f(y),y), then reflects on the other profile at the point (g(ε), ε), finally going back to source A. Clearly the variable ε depends on the variable y, so we can write ε(y). According to Fermat’s principle, a light ray trajectory between any two points must be such that the optical path length is stationary. Therefore, when two points (x_A, 0) and (g(ε), ε) are fixed, then Fermat’s principle implies that

 

Fig. 3 Nomenclature to formulate the reflector Type I for two spherical wavefronts.

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On the other side a ray leaving source B reflects on the first profile at the point (f(y),y), then reflects on the other profile at the point (g(φ), φ), finally going back to source B, giving

The arguments ε and φ are functions of y, Eq. (1)–(4) constitute a system of four functional differential equations, with unknown functions f(y), g(y), ε(y) and φ(y). The following contour conditions are set:$f(h)=g(h)=0$, $\epsilon (h)=\varphi (h)=h$, f ^¢(0) = tanα and g^¢(0) = −tanβ, where α + β = π/2.

It has been proven [5] that this class of functional differential equations has a unique analytic solution (i.e., functions f(y), g(y), ε(y) and φ(y) can be expressed as a Taylor power series about y = h) for each set of values x ₀, x ₁, h, and α. The convergence radius of the series is given by the value of y for which either ε’(y) or φ’(y) vanishes [5].

The generalization of this approach to other wavefronts WF_A and WF_B can be done by substituting functions $d_i$ and ${\widehat{d}}_i$with the eikonal functions associated with those wavefronts. The existence and uniqueness of the analytic solution is expected to be obtained provided that the functions (x_A(t), y_A(t)) and (x_B(t’), y_B(t’)) describing the caustics of those two wavefronts are also analytic functions.

3. Design procedure

The resolution of the system of Eqs. (1)–(4) is via an approximate procedure comprising two steps. The first step is calculation of a polynomial approximation of the two sides of the reflector profile near the groove peak. Once this is done, we select a segment of one side of the profile, and apply the SMS design method starting from that segment to build the entire groove reflector. The approximation error can be made as small as needed just by selecting the segment close enough to the groove peak.

3.1 Polynomial approximation near the groove peak

Let us introduce the functions:

Then Eqs. (1)–(4) can be written as F_j = 0, where j = 1,2,3,4. Since by [5] the solutions f(y), g(y), ε(y) and φ(y) are analytic about y = h, they are functions F_j(y) as well, so that by Taylor’s theorem:

Thus Eqs. (1)–(4) are fulfilled if n→∞ and

System (7) comprises 4(n + 1) nonlinear equations with 4(n + 1) unknowns, and is quite difficult to solve even for small values of n. The unknown quantities of the system are the values of the i-th derivatives of functions f(y), g(y), ε(y) and φ(y) at y = h, where i = 0,1,...,n. We have solved system (7) for n = 3, giving Taylor polynomials of degree 3 for each function. This requires an additional condition, the value of the first derivative of f(y) at y = h.

In the symmetric case, $f(y)=-g(y)$ and ε(y) = φ(y), so that system (7) can be reduced to a system of 2(n + 1) nonlinear equations with 2(n + 1) unknown quantities. Since the number of equations is lower, we have solved this symmetric case for n = 4, which leads to a Taylor polynomials of degree 4 for each function.

3.2 Calculation of the complete reflector profiles with the SMS method

In this second step, the SMS 2D method [6] is applied. This method builds a sequence of isolated points of the solution starting from the location and normal vector of a point of one side of the reflector, as shown next. Consider the scheme of a Type I reflector, shown in 
Fig. 4(a) . For given wavefronts centered at A and B, a design continues by choosing a point P₀ on the polynomial approximation of the curve (g(y),y), in a neighbourhood of the y = h obtained previously. Let n_P0 be the normal vector to the polynomial approximation at P₀. Once {P₀, n_P0} are prescribed we are able to calculate a point on the other surface Q₀. The point Q₀ is calculated along the trajectory of the ray from A, after the reflection at P₀ being the one satisfying that the total optical path length equals $L_{AA}=2\sqrt{x_A^2+h^2}$. The normal vector n_Q0 is then calculated which produces the reflection from P₀-Q₀ to Q₀-A. The procedure continues thereafter using the ray form B impinging at the point {Q₀, n_Q0}, thereby calculating the next point P₁ of the first surface (Fig. 4 the ray in blue), using the optical path length $L_{BB}=2\sqrt{x_B^2+h^2}$. This procedure is repeated to obtain further points Q₁, P₂, Q₂, etc. along the curves (with the y coordinate of the points decreasing thereby).

 

Fig. 4 SMS 2D method of reflector design: a) Type I (b) Type II.

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Other points P_-1, Q_-1, P_-2, etc. are obtained when one starts the SMS procedure using the point P₀ and the ray from B (instead of the ray from A, as before). The sequence of points {…, P_-2, Q_-1, P_-1, Q₀, P₀, Q₁, P₂, Q₂, …} together with their associated normal vectors is called an SMS chain, in the SMS nomenclature. The sequence {P_i} results converge to the groove peak for i→-∞. The sequence {P_i} for i→ + ∞ does not, however, generally converge.

Once the first SMS chain is calculated from {P₀, n_P0} we choose two consecutive points of the sequence on one side, for example, P₀ and P₁. Interpolating a C¹ line segment in between them defines a set of new initial points lying on this segment. That segment will be very close to the approximate polynomial curve (g(y),y) previously calculated, and their difference can be made as small as desired by choosing P₀ close enough to the groove peak.

By launching a set of rays from A towards the points of the segment P₀P₁ with the same optical path length, a new set of points on the other surface is calculated, between points Q₀ and Q₁ (Fig. 4, those rays indicated by red dashed lines). Repeating the same process, now using rays from B and the points of segment Q₀Q₁, we obtain the points of the segment P₁P₂. This procedure can be implemented until the all segments are filled by new points. The number of the points of the first segment P₀P₁ can be chosen without any restriction, so the density of the calculated curve points can be as high as necessary for accurate specification.

4. Results

4.1 Two spherical wavefronts

These designs have been traced in the commercial simulation package Light Tools. Figure 5 shows the ray traces of two Type I reflectors, one symmetric and the other asymmetric, both with the same optical path length and distance between the sources. Figure 6 shows two models of a Type II reflector. The symmetric solution corresponds to the parameter α = π/4 and the asymmetric solution to α≠0.

 

Fig. 5 Ray-trace simulations for a symmetric and an asymmetric Type I reflector for two spherical wavefronts.

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Fig. 6 Ray-trace simulations for a symmetric and an asymmetric Type II reflector, for two spherical wavefronts.

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4.2 Circular caustic

Microstructures based on small V-groove reflectors are already in use in solar collectors. These structures are designed to guarantee efficient transmission from the sun to the circular receiver and to provide a separation between reflector and receiver [7–9]. Figure 7 shows a different type of V-reflector designed to transfer all rays emanating from a circular source back to itself. Due to the edge ray theorem all source edge rays must be sent back on to themselves. This is a case of Type I reflector design in which the wavefronts WF_A and WF_B define a common circular caustic. Then, the functions (x_A(t), y_A(t)) and (x_B(t’), y_B(t’)) are two different parameterizations of the circle.

 

Fig. 7 Ray-trace simulation of a symmetric V-reflector design for a circular caustic.

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5. Summary

The SMS method has been shown to be a very powerful method for aspheric V-reflector designs. Taylor expansion of the reflector surfaces near the convergent points (groove peaks) provides a good approximation of the first curve segment and improves the smoothness of the surfaces. All the canonical examples presented show perfect coupling of two wavefronts.