## Abstract

The Simultaneous Multiple Surface design method in three dimensions (SMS3D) is applied to the design of free-form V-groove reflectors. The general design problem is how to achieve the coupling of two wavefronts after two reflections at the V-groove, no matter which side of the groove the rays hit first. This paper also explains a design procedure for thin dielectric grooved-reflector substitutes for conventional mirrored surfaces. Some canonical V-groove designs are ray-traced in detail.

## 1. Introduction

The linear V-groove reflector is a well known optical device, and its use has spread to several applications, particularly signaling and displays.

This linear reflector is formed by two flat profiles that orthogonally join at a straight groove edge-line. If that line is parallel to the x axis (see Fig. 1 ), an incident ray with direction cosines (p,q,r) reflects at each of the two sides of the reflector, ending up with direction cosines (p,-q,-r).

Fig. 1 The linear 90° groove reflector 3D view and its front view.

The design of non-flat V-groove reflectors in three dimensions has already been considered in [68] for the particular design problem of coupling between a plane and a spherical wavefront, so that the V-groove performs the same function as a parabolic mirror. The design procedure in [7] and [8] approximates the groove’s cross-sectional profile with straight lines, while the method use in [6] is based on numerical optimization.

In this paper we examine the more general design problem of coupling two arbitrary given wavefronts, and derive formally exact solutions with a direct method (i.e., not by numerical optimization) based on the Simultaneous Multiple Design method in three dimensions (SMS3D) [3,4]. These solutions comprise two intersecting free-form surfaces (i.e., no linear or rotational symmetry), as illustrated in Fig. 2 . In Section 2 we formulate the problem and study the existence and uniqueness of solutions defined by a set of analytic functions (i.e., admitting a series expansion). In Section 3, a constructive design procedure is disclosed. Sections 4 and 5 give examples and description of the applications.

Fig. 2 Free-form V-groove reflector.

The SMS design method has already been applied to the design of aspheric V-groove reflector profiles (i.e., in only two dimensions, SMS2D) [2]. The free-form designs described here are an extension to 3D of the Type II 2D reflector that was explained in [2].

## 2. Statement of the problem

A V-groove free-form reflector is tasked to couple the rays of two wavefronts, spherical WFA and planar WFB, after consecutive reflections on each side of the groove, either one first. At the groove edge-line, both reflections are at the same point (as one of the rays does in Fig. 2). In some cases, the profiles can be symmetric with respect to the groove edge-line but, in general, the profiles are asymmetric.

Without loss of generality Fig. 3 illustrates the overall design problem, showing the case of a plane wavefront with ray vectors v o and v o′ and a spherical wavefront with its center shown as originating the ray vectors. The two free-form surfaces P(u,v) and Q(u,v) form the V-groove reflector. The parameterization of both surfaces is preferably done in an interrelated manner, as follows. Parameter v is the arc length along the groove edge-line G(v), along which the groove walls intersect to define the bottom of the groove. (In actual implementation, rounding, or filleting, of this groove is inevitable but it can be kept too small for the human eye to discern.) The parameter u for each surface coincides with the arc length of two specific v = constant lines (shown in green). These two lines are contained in the V-groove surfaces and intersect at the groove edge-line G(v). Those lines fulfil two conditions: they are analytic, and after a first reflection at either line there will be another reflection at the other line. Those conditions unequivocally define the v = constant lines, under certain contour conditions, as proven in Section 3.1. Figure 3a shows two v = constant lines (the lines in green) and two representative rays for the second condition, which hit the surfaces on points P(u,v), Q(u′,v) and Q(u′′,v). Note that the variables u′ and u′′ depend on the variable u, so we can write u′(u) and u′′(u). The normal vectors of the surfaces, described by functions NP(u,v) and NQ(u,v), are not shown in Fig. 3a.

Fig. 3 (a) In green, ν = constant lines of the V-groove surfaces are such that after a first reflection at either line the second reflection will be at the other line. (b) Vectors at a point of the groove edge-line (G)(v) and contained in the plane perpendicular to its tangent vector t = (G)′(v). The angle α(v) is defined as the one formed by the normal vector of the surfaces (P), NP(0,v), and the plane formed by the ray vectors v i and v o impinging on (G)(v).

Straightforward but dense calculations that are detailed in the Appendix yield:

• 1. The groove edge line G(v) is defined as a solution of the differential equation:

$(vo+vi)×G'(v)=0.$

This Eq. (1) means that v o + v i is parallel to G′( v ), where G′( v ) is the tangent vector of the groove edge-line, and v i and v o are the ray vectors of WFA and WFB, respectively, passing through G(v). In the example considered in Fig. 3, the center of WFA is the origin of coordinates, so that v i = G(v)/│G(v)│ and v o = (1,0,0).

• 2. As in the case of the linear V-groove reflector of Fig. 1, the normal vectors to the surfaces, NP(0,v) and NQ(0,v) on the groove edge line G(v) are mutually perpendicular (see Fig. 3b).
• 3. The V-groove surfaces, P(u,v) and Q(u,v), together with normal vectors NP(u,v), NQ(u,v), and the auxiliary functions u′(u) and u′′(u), are defined as a solution of a system of ordinary functional differential equations (Eq. (8)-(14) in the Appendix) in the variable u for each v. As shown in the Appendix, there exists a unique analytic solution, provided that G(v) and α(v) are known (see Fig. 3.b) and there also exists a unique second order approximation of the unknown functions.

## 3. Design procedure

The construction of the system of Eq. (8)-(14) is by a procedure comprising nine Steps:

• Step 1. Calculate the groove edge line G(v) by solving differential Eq. (1) with the contour condition G(0) = G 0 (G 0 can be chosen freely). Then it is possible to calculate the optical path length L between the two wavefronts.
• Step 2. Set v = 0 and consider the value α(0) = α 0. This value can be chosen freely, as well.
• Step 3. Calculate a second-order approximation in the variable u (for v = constant) of the unknown functions P(u,v), Q(u,v), NP(u,v), NQ(u,v), u′(u) and u′′(u) of the groove surfaces near the groove edge line, for a fixed value of v. As shown in the Appendix, this is obtained by solving the Taylor series expansion of Eq. (8)-(14).
• Step 4. Apply the SMS3D design method [3] starting from one point of one of the approximated v = constant lines, designated as P 0, and the normal vector to the surface there (NP 0). The method builds the entire lengths of the v = constant lines P(u,v), Q(u,v) of the V-groove reflector, for the desired range u = 0 to u = umax(v). In that calculation, the normal vector of the surfaces on those lines, functions NP(u,v), NQ(u,v), and the function linking the parameters u′(u) and u′′(u) are obtained, as well. Details are in Section 3.1
• Step 5. Compute the partial-derivative functions P u(u,v) and NP u(u,v)
• Step 6. Consider an incremental increase Δv in the value of parameter v, and the corresponding point G(v + Δv) on the groove line. Initially, take α(v + Δv) = α(v).
• Step 7. Repeat Steps 3 to 5 for the new value v + Δv in order to calculate functions P(u,v + Δv) and NP(u,v + Δv). Compute the integrability condition on surface P around v = constant line P(u,v), using the Malus-Dupin theorem [5] in the form:

$e(u,v+Δv)=Pu(u,v)⋅NP(v+Δv)−NP(v)Δv−P(v+Δv)−P(v)Δv⋅NPu(u,v).$

• Step 8. Repeat Step 7, iterating on the value of α(v + Δv) to minimize:

$E(v)=1umax(v)∫u=0umax(v)e2du.$

If the minimum of E(v) is exactly zero, so will e(u,v + Δv), and surface P(u,v) will then be integrable. We have not theoretically proven that the minimum of E(v) will be zero, but that seems to be the case in all the examples calculated. Note that if surface P(u,v) is integrable, Q(u,v) will be automatically integrable, since both are linked by the invariance of the optical path length between the wavefronts.

• Step 9. Increment the v value again and repeat steps 3 to 7 to advance to the next point G(v + 2Δv) of the groove edge-line.

#### 3.1 SMS3D calculation

Step 4 in the previous section applies the SMS3D method to calculate the v = constant lines from a starting point on one of the surfaces, given its normal vector, designate them P 0 and NP 0. This is an extension to 3D of the SMS2D V-groove construction explained in [2], in which the 3D lines are constructed point-by-point. Each new point of one of the lines permits the calculation of a further point of the other line, and so on [1].

Without loss of generality, the SMS3D calculation can be explained using the example shown in Fig. 3a, which provides coupling of a plane and a spherical wavefront. In this case, the vector field of the input and output rays are defined as v i(r) = r/│r│ and v o(r) = z, where z = (0,0,1) in Cartesian coordinates. It is easy to see that the groove edge-lines (calculated using Eq. (1)) are parabolas with focus at the origin and axis coincident with the axis z. Choose one of those parabolas as the groove edge line.

Figure 4 shows the V-groove reflector and the v = constant lines (in green) to be calculated, which were also shown in Fig. 3a. The point P 0, calculated in Step 3 in the neighborhood of the groove edge-line, has not been drawn very close to that line in Fig. 4 for clarity. The SMS3D design will first involve the calculation of a sequence of points on both surfaces (called an SMS chain [3]), shown in Fig. 4. From the starting point P 0 and NP 0, on surface P(u,v), the next point Q 0 on the v = constant line of surface Q(u,v) is calculated along the trajectory of the ray from WFA, after the reflection at {P 0, NP 0} being the one with a total optical path length equalling L, which was calculated in Step 1. The normal vector NQ 0 is then calculated that produce the reflection from P 0-Q 0 to Q 0-WFB. Then, the procedure continues thereafter using the ray from WFA impinging at the point {Q 0, NQ 0}, thereby calculating the next point P 1 on the v = constant line of the surface P(u,v). This procedure is repeated to obtain further points Q 1, P 2, Q 2, etc. along the lines. Note that the points of the sequence {P 0, Q 0, P 1, Q 1, P 2, Q 2, } are not, in general, coplanar.

Fig. 4 3D V- groove reflector which reflects a plane into a spherical wavefront. Perspective and front views.

An SMS skinning process [3] calculates the remaining points of the v = constant lines. Consider an interpolation between two adjacent points {P 0, NP 0} and {P 1, NP 1}. Each point of the interpolated line is taken as initial point of a further sequence of points. The interpolation is taken from the second-order approximation of Step 3, in the same way that was carried out in the 2D V-groove designs in [2].

## 4. Results

Two V-groove designs providing the coupling between a plane and a spherical wavefront have been generated using the previous procedure. The first one, shown in Fig. 4, is a symmetric design for which the function α(v) = π/4 (i.e, 45°) is constant along the groove edge-line. In the second design (Fig. 5 ), an asymmetric contour condition α(v) = π/6 (i.e, 30°) has been forced. In this case, instead of using the procedure explained in Section 3 to find the function α(v), we tried directly the constant function α(v) = π/6 and build the surfaces. Then, by checking the integrability condition and performing ray traces we have confirmed that such a guess seems to be right.

Fig. 5 LightTools simulation for an asymmetric 3D V-groove design for a plane and a spherical wavefront.

Both V-grooves have been simulated in the Synopsys software package LightTools [10] by ray-tracing parallel rays in a 400x200 mm rectangle (the rays from WFB) towards the groove. A flat receiver was positioned parallel to WFB and passing through the center C of spherical wavefront WFA. In the case of perfect coupling, all the rays would impinge on the point C. Ray-tracing results show that, in both cases, more than 90% of the rays have reached the receiver plane inside the 10x10 μm square centered with the spherical wavefront, while the other 10% are inside the 30x30 μm square.

Figure 6 shows another canonical example: a reflector that achieves coupling of two spherical wavefronts. In this case the groove edge-lines are ellipses having focus at the centers of the spherical wavefronts. The selected design is symmetric (α(v) = π/4) and has been simulated in the LightTools as well, giving similar high efficiency as the previous design.

Fig. 6 LightTools simulation for the symmetric 3D V-groove design coupling two spherical wavefronts.

## 5 . Applications

We also provide a design procedure for a thin dielectric free-form reflector providing coupling between two arbitrary wavefronts (Fig. 7 ). The entrance surface is the smooth side while the grooved surface comprises an array of free-form grooves. The conventional free-form reflector that performs this function is the reflective generalised Cartesian Oval, discovered by Levi-Civita in 1900 [5].

Fig. 7 Free from thin dielectric sheet that acts as a TIR reflector.

The design starts by prescribing the form of the entrance surface, which can be rather arbitrary. However, a particularly preferred entrance surface is one that coincides with a (reflective) generalised Cartesian oval, that couples WF1 and WF2, which will increase the efficiency of this mirroless reflector by coupling the Fresnel reflection at that entrance surface. Then, we refract wavefronts WF1 and WF2 through the entrance surface to obtain a new pair of wavefronts WFA and WFB. For two given wavefronts WFA and WFB, using Eq. (1), one can calculate the candidates for the groove edge-lines. Once the edge-lines are selected, the grooves are built as described in Section 3.

This design procedure does not ensure that the rays coming to the groove surface will suffer two TIR reflections. This condition has to be independently checked after the reflector has been calculated. If, however, the reflector is thin, the grooves will be small and their shape in the normal planes to the edge groove line will be close to a flat 90° corner.

When the normal sections along the groove are a 90° corner, the TIR condition is fulfilled for all rays with a (p,q) representation inside the solid region of Fig. 8 (for the refractive index n = 1.492). Now (p,q,r) represents cosines respect to the vectors t, n 1 and n 2, where t is tangent to the groove edge-line. The orthogonal system formed by the vectors t, n 1 and n 2 changes along the edge line. The narrower portion of the solid region corresponds to rays with p = 0, i.e., rays contained in planes normal to the groove edge-line. For these rays q must be between ± {(n 2-1)½-1}/2½, which implies that n must be greater than 2½ = 1.414 for a non-null range of q to exist at p = 0. For rays with p≠0, the range of q for which two total internal reflections are achieved is bigger.

Fig. 8 V-groove reflector having a 90° corner as normal sections. Condition for rays having two TIR at the groove sides.

## 6. Conclusions

The SMS3D method has been shown to be a very powerful method for free-form V-reflector designs. All the canonical examples presented show excellent coupling of two wavefronts. The presented design procedure is suitable for calculation of mirrorless TIR reflectors. These reflectors will be a capable alternative to conventional metallized reflectors, especially because of the potential for mass production, by injection molding, which reduces their cost.

## Appendix

#### Conditions on the groove edge-line

Consider the v = constant lines on surfaces P(u,v) and Q(u,v) shown in Fig. 3a. We will omit the v dependence of the functions P(u,v), Q(u,v), NP(u,v) and NQ(u,v) in order to simplify our nomenclature, but it will be shown explicitly when needed to avoid confusion.

Let v i be the ray vector for a ray coming from spherical wavefront WFA in Fig. 3a and impinging on the groove point P(u,v). This ray is reflected twice, once at each v = constant line, at the points P(u,v) and Q(u,v). Clearly the variable u′ depends on the variable u, as expressed by the function u′(u). The reflection law for these two reflections gives: v x=v i-2(v i ·NP(u))NP(u) and v o=v x-2(v x ·NQ(u′))NQ(u′), where NP(u) and NQ(u′) are the normal vectors of the surfaces at P(u) and Q(u′), v o is reflected ray vector, and v x is the ray vector after the first reflection (Fig. 3a). Eliminating v x from these equations yields the two-reflection law at the groove sides:

$vo=vi−2(vi⋅NP(u))NP(u)−2[vi⋅NQ(u′)−2(vi⋅NP(u))NP(u)⋅NQ(u′)]NQ(u′).$

Also, for a ray leaving WFA that reflects first at the point Q(u′′(u)), then reflects on the other side at P(u) and then goes to WFB, we similarly have: v x′ = v i′-2(v i·NQ(u′′))NQ(u′′) and v o′ = v x′-2(v x· NP(u))NP(u). Eliminating v x′ from these equations yields

$v′o=v′i−2(v′i⋅NQ(u″))NQ(u″)−2[v′i⋅NP(u)−2(v′i⋅NQ(u″))NQ(u″)⋅NP(u)]NP(u).$

At the groove edge-line, both reflections occur at the same point, so that v i = v i′ and v o = v o′ and if we subtract Eq. (5) from Eq. (4) with u = 0, we get

$(NP(0)⋅NQ(0))[(vi⋅NP(0))NQ(0)−(vi⋅NQ(0))NP(0)]=0.$

This means that NP(0)⊥NQ(0), so the lines P(u) and Q(u) form a 90° corner at the groove edge-line. The second solution of Eq. (6), NP(0) = NQ(0) represents the conventional ungrooved single-surface reflector. Imposing the condition NP(0)⊥NQ(0) upon Eq. (4) and Eq. (5) yields for u = 0 the collapse of the equations to the same form: v o = v i-2(v i ·NP(0))NP(0)-2(v i ·NQ(0))NQ(0). Since any vector v i can be written in terms of its components on the triorthogonal system NP(0), NQ(0) and t as v i = (NP(0)·v i)NP(0) + (NQ(0)·v i)NQ(0) + (t·v i)t (where t = G’(v) is the tangent vector to the groove edge line), the two-reflection law for grooved surfaces at the groove edge-line is

$vo+vi=2(vi⋅t)t.$

At an arbitrary point the vectors v o and v i are known (the rays coming from the wavefronts WFA and WFB), thus from Eq. (7) we see that v o + v i is parallel to t, as stated in Eq. (1), which is the equation for that vector field. Integrating it provides the candidate lines for groove edge-lines.

Note: we have calculated that NP(0)⊥NQ(0), but we do not know the exact position of these vectors in the plane perpendicular to the tangent vector t. The orientation of the vectors NP(0) and NQ(0) is defined by the angle α(v) as shown in Fig. 3b.

#### Functional differential equations of the v = constant lines

Once the groove edge-line G(v) is selected as a candidate, the v = constant lines are calculated starting at the points of G(v). These lines are given as P(u,vc,α(vc)) and Q(u,vc,α(vc)), where vc, is a value of the arc-length parameter v. The normal vectors of each grooved surface along these lines are given as NP(u,vc,α(vc)) and NQ(u,vc,α(vc)).

Let us now find the equations defining v = contant lines P(u) and Q(u). Consider first the ray WFA-P(u)-Q(u′)-WFB (the ray in blue, Fig. 3a). The first reflection at P(u) is given by v x = v i-2(v i ·NP(u))NP(u), which means that vectors v i-v x and NP(u) have the same direction. Therefore, v i-v x has to be perpendicular to the vectors P’(u) (where the prime denotes the partial derivative of P u(u,v)) and P’(u) × NP(u), where P’(u) is tangent vector to the line at P(u), so that

$(vi−vx)⋅P′(u)=0, (vi−vx)⋅(P′(u)×NP(u))=0.$

The second reflection at the point Q(u′) is given by v o = v x-2(v x ·NQ(u′))NQ(u′), so that v x-v o must be perpendicular to the vectors Q’(u′) and Q’(u′) × NQ(u′), where Q’(u′) is tangent vector to the line at Q(u′), hence

$(vx−vo)⋅Q′(u′)=0, (vx−vo)⋅(Q′(u′)×NQ(u′))=0.$

Since u′(u) is an unknown function, the equations in Eq. (9) are functional differential equations.

Similar equations can be established for the ray WFA-Q(u′′)- P(u)-WFB (the ray in red, Fig. 3a). For this ray, the two reflections at the groove lines are given by

$(v′i−v′x)⋅Q′(u″)=0, (v′i−v′x)⋅(Q′(u″)×NQ(u″))=0,$
$(v′x−v′o)⋅P′(u)=0, (v′x−v′o)⋅(P′(u)×NP(u))=0.$

Here Q’(u′′) is the tangent vector to the line at Q(u′′). Having in mind that the parameter u respresents the line’s arc-length, we have

$|P′(u)|=1, |Q′(u)|=1.$

Also we need to make sure that NP(u) ⊥ P’(u) and NQ(u) ⊥ Q’(u) so that

$NP(u)⋅P′(u)=0, NQ(u)⋅Q′(u)=0.$

Finally, the fact that the normals to the lines are unit vectors gives

$|NP(u)|=1, |NQ(u)|=1.$

For each value of v, the system Eq. (8)-(14) comprises fourteen equations (some of them are functional differential equations, for example Eq. (9)), with fourteen unknown functions P(u), Q(u), NP(u), NQ(u), u′(u) and u′′(u). Also, since NP(0)⊥NQ(0), these vectors are in the plane perpendicular to t and they are defined by α(v), Fig. 3b.

Assuming α(v) and G(v) are given, this system of functional differential equations belong to the class described in [9], wherein it is proven that there exists a unique analytic solution provided that (1) the following contour conditions are set P(0) = Q(0) = G(v), u ¢(0) = 0 and u′′(0) = 0, and (2) a unique solution of the second-order approximation of the Eq. (8)-(14) exists. This latter condition is discussed next.

#### Second-order approximation

Let us represent Eq. (8)-(14) by functions Fj(u), where j = 1,2,..14, for example F1(u)≡(v i-v x)·P′(u), F2(u)≡(v i-v x)·(P′(u) × NP(u)), ... F14(u)≡│NQ(u)│-1. Hence we have Fj(u) = 0, for each j. When the functions P(u), Q(u), NP(u), NQ(u), u′(u) and u′′(u) are analytic about u = 0, then the functions Fj(u) are analytic as well, so that by Taylor’s theorem:

$Fj(u)=Fj(0)+F′j(0)u+...+Fj(n)(0)n!un+O(un+1).$

Thus Eq. (8)-(14) are fulfilled for n→∞ and

$Fj(i)(h)=0.$

Here j = 1,2,..14 and i = 0,1,...,n.

The system in Eq. (16) comprises 14(n + 1) nonlinear equations with 14(n + 1) unknowns. The unknown quantities of the system are the values of the i-th derivatives of functions P, Q, NP, NQ, u′(u) and u′′(u) respect to u at u = 0, where i = 0,1,...,n. We have solved system in Eq. (16) for n = 2, obtaining polynomials of degree 2 as approximations for each function.

## Acknowledgments

Authors thank the Spanish Ministries MCINN (ENGINEERING METAMATERIALS: CSD2008-00066, DEFFIO: TEC2008-03773, SIGMASOLES: PSS-440000-2009-30), MITYC (ECOLUX: TSI-020100-2010-1131, SEM: TSI-020302-2010-65), the Madrid Regional Government (SPIR: 50/2010O.23/12/09, TIC2010 and O-PRO: PIE/209/2010) and UPM (Q090935C59) for the support given in the preparation of the paper. The authors thank Synopsys (formerly Optical Research Associates) for granting us the LightTools university license. The final edit of this paper was by Bill Parkyn.

1. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier Academic Press, 2004)

2. D. Grabovickić, P. Benítez, and J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18(3), 2515–2521 (2010). [CrossRef]   [PubMed]

3. P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999). [CrossRef]

4. International Patent Pending, US2010/002320 A1

5. T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).

6. L. D. DiDomenico, “Non-imaging facet based optics,” U.S. Patent US 7697219 (10 Jul 2008).

7. M. O'Neill, “Analytical and Experimental Study of Total Internal Reflection Prismatic Panels for Solar Energy Concentrators,” Technical Report No. D50000/TR 76–06, E-Systems, Inc., P.O. Box 6118, (1976)

8. A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977). [CrossRef]

9. B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).

10. Synopsys software package LightTools, http://www.opticalres.com/

### References

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1. R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier Academic Press, 2004)
2. D. Grabovickić, P. Benítez, and J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18(3), 2515–2521 (2010).
[CrossRef] [PubMed]
3. P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]
4. International Patent Pending, US2010/002320 A1
5. T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).
6. L. D. DiDomenico, “Non-imaging facet based optics,” U.S. Patent US 7697219 (10 Jul 2008).
7. M. O'Neill, “Analytical and Experimental Study of Total Internal Reflection Prismatic Panels for Solar Energy Concentrators,” Technical Report No. D50000/TR 76–06, E-Systems, Inc., P.O. Box 6118, (1976)
8. A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977).
[CrossRef]
9. B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).
10. Synopsys software package LightTools, http://www.opticalres.com/

#### 1999 (1)

P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]

#### 1993 (1)

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).

#### 1977 (1)

A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977).
[CrossRef]

#### 1900 (1)

T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).

#### Benítez, P.

P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]

#### Levi-Civita, T.

T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).

#### Miñano, J. C.

P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]

#### Mohedano, R.

P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]

#### Rabl, A.

A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977).
[CrossRef]

#### van-Brunt, B.

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).

#### Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. (1)

T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).

#### NZ J. Mathematics (1)

B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).

#### Proc. SPIE (1)

P. Benítez, R. Mohedano, and J. C. Miñano, “Design in 3D geometry with the simultaneous multiple surface design method of nonimaging optics,” Proc. SPIE 3781, 12–21 (1999).
[CrossRef]

#### Sol. Energy (1)

A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977).
[CrossRef]

#### Other (5)

R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier Academic Press, 2004)

Synopsys software package LightTools, http://www.opticalres.com/

International Patent Pending, US2010/002320 A1

L. D. DiDomenico, “Non-imaging facet based optics,” U.S. Patent US 7697219 (10 Jul 2008).

M. O'Neill, “Analytical and Experimental Study of Total Internal Reflection Prismatic Panels for Solar Energy Concentrators,” Technical Report No. D50000/TR 76–06, E-Systems, Inc., P.O. Box 6118, (1976)

### Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

### Figures (8)

Fig. 1

The linear 90° groove reflector 3D view and its front view.

Fig. 2

Free-form V-groove reflector.

Fig. 3

(a) In green, ν = constant lines of the V-groove surfaces are such that after a first reflection at either line the second reflection will be at the other line. (b) Vectors at a point of the groove edge-line (G)(v) and contained in the plane perpendicular to its tangent vector t = (G)′(v). The angle α(v) is defined as the one formed by the normal vector of the surfaces (P), NP (0,v), and the plane formed by the ray vectors v i and v o impinging on (G)(v).

Fig. 4

3D V- groove reflector which reflects a plane into a spherical wavefront. Perspective and front views.

Fig. 5

LightTools simulation for an asymmetric 3D V-groove design for a plane and a spherical wavefront.

Fig. 6

LightTools simulation for the symmetric 3D V-groove design coupling two spherical wavefronts.

Fig. 7

Free from thin dielectric sheet that acts as a TIR reflector.

Fig. 8

V-groove reflector having a 90° corner as normal sections. Condition for rays having two TIR at the groove sides.

### Equations (16)

$( v o + v i ) × G '( v ) = 0.$
$e ( u , v + Δ v ) = P u ( u , v ) ⋅ N P ( v + Δ v ) − N P ( v ) Δ v − P ( v + Δ v ) − P ( v ) Δ v ⋅ N P u ( u , v ) .$
$E ( v ) = 1 u max ( v ) ∫ u = 0 u max ( v ) e 2 d u .$
$v o = v i − 2 ( v i ⋅ N P ( u ) ) N P ( u ) − 2 [ v i ⋅ N Q ( u ′ ) − 2 ( v i ⋅ N P ( u ) ) N P ( u ) ⋅ N Q ( u ′ ) ] N Q ( u ′ ) .$
$v ′ o = v ′ i − 2 ( v ′ i ⋅ N Q ( u ″ ) ) N Q ( u ″ ) − 2 [ v ′ i ⋅ N P ( u ) − 2 ( v ′ i ⋅ N Q ( u ″ ) ) N Q ( u ″ ) ⋅ N P ( u ) ] N P ( u ) .$
$( N P ( 0 ) ⋅ N Q ( 0 ) ) [ ( v i ⋅ N P ( 0 ) ) N Q ( 0 ) − ( v i ⋅ N Q ( 0 ) ) N P ( 0 ) ] = 0.$
$v o + v i = 2 ( v i ⋅ t ) t .$
$( v i − v x ) ⋅ P ′ ( u ) = 0 , ( v i − v x ) ⋅ ( P ′ ( u ) × N P ( u ) ) = 0.$
$( v x − v o ) ⋅ Q ′ ( u ′ ) = 0 , ( v x − v o ) ⋅ ( Q ′ ( u ′ ) × N Q ( u ′ ) ) = 0.$
$( v ′ i − v ′ x ) ⋅ Q ′ ( u ″ ) = 0 , ( v ′ i − v ′ x ) ⋅ ( Q ′ ( u ″ ) × N Q ( u ″ ) ) = 0 ,$
$( v ′ x − v ′ o ) ⋅ P ′ ( u ) = 0 , ( v ′ x − v ′ o ) ⋅ ( P ′ ( u ) × N P ( u ) ) = 0.$
$| P ′ ( u ) | = 1 , | Q ′ ( u ) | = 1.$
$N P ( u ) ⋅ P ′ ( u ) = 0 , N Q ( u ) ⋅ Q ′ ( u ) = 0.$
$| N P ( u ) | = 1 , | N Q ( u ) | = 1.$
$F j ( u ) = F j ( 0 ) + F ′ j ( 0 ) u + ... + F j ( n ) ( 0 ) n ! u n + O ( u n + 1 ) .$
$F j ( i ) ( h ) = 0.$