## Free-form V-groove reflector design with the SMS method in three dimensions |

Optics Express, Vol. 19, Issue S4, pp. A747-A756 (2011)

http://dx.doi.org/10.1364/OE.19.00A747

Acrobat PDF (1294 KB)

### 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.

© 2011 OSA

## 1. Introduction

*p*,

*q*,

*r*) reflects at each of the two sides of the reflector, ending up with direction cosines (

*p*,

*-q*,

*-r*).

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

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

*i.e.*, not by numerical optimization) based on the Simultaneous Multiple Design method in three dimensions (SMS3D) [3

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]

*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.

*i.e.*, in only two dimensions, SMS2D) [2

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]

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]

## 2. Statement of the problem

**and planar**

*WF*_{A}**, 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.**

*WF*_{B}**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

**and**

*WF*_{A}**, respectively, passing through**

*WF*_{B}**G**(

*v*). In the example considered in Fig. 3, the center of

**is the origin of coordinates, so that**

*WF*_{A}**v**

_{i}=

**G**(

*v*)/│

**G**(

*v*)│ and

**v**

_{o}= (1,0,0).

- 3. The V-groove surfaces,
**P**(*u,v*) and**Q**(*u,v*), together with normal vectors**N**(_{P}*u,v*),**N**(_{Q}*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

*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*),**N**(_{P}*u,v*),**N**(_{Q}*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 approximated3. 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]*v*= constant lines, designated as**P**_{0}, and the normal vector to the surface there (**N**_{P}_{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*=*u*(_{max}*v*). In that calculation, the normal vector of the surfaces on those lines, functions**N**(_{P}*u,v*),**N**(_{Q}*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**N**_{P}(_{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**N**(_{P}*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:*Step 8.*Repeat Step 7, iterating on the value of*α*(*v*+*Δv*) to minimize:

*E*(

*v*) is exactly zero, so will

*e*(

*u*,

*v +*Δ

*v*), 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

*v*= constant lines from a starting point on one of the surfaces, given its normal vector, designate them

**P**

_{0}and

**N**

_{P}_{0}. This is an extension to 3D of the SMS2D V-groove construction explained in [2

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]

**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.

*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

**3781**, 12–21 (1999). [CrossRef]

**P**

_{0}and

**N**

_{P}_{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

**, after the reflection at {**

*WF*_{A}**P**

_{0},

**N**

_{P}_{0}} being the one with a total optical path length equalling

*L*, which was calculated in Step 1. The normal vector

**N**

_{Q}_{0}is then calculated that produce the reflection from

**P**

_{0}-

**Q**

_{0}to

**Q**

_{0}-

**. Then, the procedure continues thereafter using the ray from**

*WF*_{B}**impinging at the point {**

*WF*_{A}**Q**

_{0},

**N**

_{Q}_{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.

**3781**, 12–21 (1999). [CrossRef]

*v*= constant lines. Consider an interpolation between two adjacent points {

**P**

_{0},

**N**

_{P}_{0}} and {

**P**

_{1},

**N**

_{P}_{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

**18**(3), 2515–2521 (2010). [CrossRef] [PubMed]

## 4. Results

*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.

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

*mm*rectangle (the rays from

**) towards the groove. A flat receiver was positioned parallel to**

*WF*_{B}*WF*and passing through the center

_{B}**C**of spherical wavefront

**. In the case of perfect coupling, all the rays would impinge on the point**

*WF*_{A}**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.

*α*(

*v*) =

*π*/4) and has been simulated in the LightTools as well, giving similar high efficiency as the previous design.

## 5 . Applications

**and**

*WF*_{1}**, which will increase the efficiency of this mirroless reflector by coupling the Fresnel reflection at that entrance surface. Then, we refract wavefronts**

*WF*_{2}**and**

*WF*_{1}**through the entrance surface to obtain a new pair of wavefronts**

*WF*_{2}**and**

*WF*_{A}**. For two given wavefronts**

*WF*_{B}**and**

*WF*_{A}**, 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.**

*WF*_{B}*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}/2

^{½}, which implies that

*n*must be greater than 2

^{½}= 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.

## 6. Conclusions

## Appendix

### Conditions on the groove edge-line

*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*),

**N**(

_{P}*u,v*) and

**N**(

_{Q}*u,v*) in order to simplify our nomenclature, but it will be shown explicitly when needed to avoid confusion.

**v**

_{i}be the ray vector for a ray coming from spherical wavefront

**in Fig. 3a and impinging on the groove point**

*WF*_{A}**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}

**·N**(

_{P}*u*))

**N**(

_{P}*u*) and

**v**

_{o}=

**v**

_{x}-2(

**v**

_{x}

**·N**(

_{Q}*u*′))

**N**(

_{Q}*u*′), where

**N**(

_{P}*u*) and

**N**(

_{Q}*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:

**that reflects first at the point**

*WF*_{A}**Q**(

*u*′′(

*u*)), then reflects on the other side at

**P**(

*u*) and then goes to

**, we similarly have:**

*WF*_{B}**v**

_{x}′ =

**v**

_{i}′-2(

**v**

_{i}′

**·N**(

_{Q}*u*′′))

**N**(

_{Q}*u*′′) and

**v**

_{o}′ =

**v**

_{x}′-2(

**v**

_{x}′

**· N**(

_{P}*u*))

**N**(

_{P}*u*). Eliminating

**v**

_{x}′ from these equations yields

**v**

_{i}=

**v**

_{i}′ and

**v**

_{o}=

**v**

_{o}′ and if we subtract Eq. (5) from Eq. (4) with

*u*= 0, we get

**N**(0)⊥

_{P}**N**(0), so the lines

_{Q}**P**(

*u*) and

**Q**(

*u*) form a 90° corner at the groove edge-line. The second solution of Eq. (6),

**N**(0) =

_{P}**N**(0) represents the conventional ungrooved single-surface reflector. Imposing the condition

_{Q}**N**(0)⊥

_{P}**N**(0) upon Eq. (4) and Eq. (5) yields for

_{Q}*u*= 0 the collapse of the equations to the same form:

**v**

_{o}=

**v**

_{i}-2(

**v**

_{i}

**·N**(0))

_{P}**N**(0)-2(

_{P}**v**

_{i}

**·N**(0))

_{Q}**N**(0). Since any vector

_{Q}**v**

_{i}can be written in terms of its components on the triorthogonal system

**N**(0),

_{P}**N**(0) and

_{Q}**t**as

**v**

_{i}= (

**N**(0)

_{P}**·v**

_{i})

**N**(0) + (

_{P}**N**(0)

_{Q}**·v**

_{i})

**N**(0) + (

_{Q}**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

**v**

_{o}and

**v**

_{i}are known (the rays coming from the wavefronts

**and**

*WF*_{A}**), thus from Eq. (7) we see that**

*WF*_{B}**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.

**N**(0)⊥

_{P}**N**(0), but we do not know the exact position of these vectors in the plane perpendicular to the tangent vector

_{Q}**t.**The orientation of the vectors

**N**(0) and

_{P}**N**(0) is defined by the angle α(

_{Q}*v*) as shown in Fig. 3b.

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

**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,v*,α(

_{c}*v*)) and

_{c}**Q**(

*u,v*,α(

_{c}*v*)), where

_{c}*v*, is a value of the arc-length parameter

_{c}*v*. The normal vectors of each grooved surface along these lines are given as

**N**(

_{P}*u,v*,α(

_{c}*v*)) and

_{c}**N**(

_{Q}*u,v*,α(

_{c}*v*)).

_{c}**P**(

*u*) and

**Q**(

*u*). Consider first the ray

**-**

*WF*_{A}**P**(

*u*)

**-Q**(

*u*′)-

**(the ray in blue, Fig. 3a). The first reflection at**

*WF*_{B}**P**(

*u*) is given by

**v**

_{x}=

**v**

_{i}-2(

**v**

_{i}

**·N**(

_{P}*u*))

**N**(

_{P}*u*), which means that vectors

**v**

_{i}-

**v**

_{x}and

**N**(

_{P}*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*) ×

**N**(

_{P}*u*), where

**P’**(

*u*) is tangent vector to the line at

**P**(

*u*), so that

**Q**(

*u*′) is given by

**v**

_{o}=

**v**

_{x}-2(

**v**

_{x}

**·N**(

_{Q}*u*′))

**N**(

_{Q}*u*′), so that

**v**

_{x}-

**v**

_{o}must be perpendicular to the vectors

**Q’**(

*u*′) and

**Q’**(

*u*′) ×

**N**(

_{Q}*u*′), where

**Q’**(

*u*′) is tangent vector to the line at

**Q**(

*u*′), hence

**-**

*WF*_{A}**Q**(

*u*′′)-

**P**(

*u*)-

**(the ray in red, Fig. 3a). For this ray, the two reflections at the groove lines are given by**

*WF*_{B}**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

**N**(

_{P}*u*) ⊥

**P’**(

*u*) and

**N**(

_{Q}*u*) ⊥

**Q’**(

*u*) so that

*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*),

**N**(

_{P}*u*),

**N**(

_{Q}*u*),

*u*′(

*u*) and

*u*′′(

*u*). Also, since

**N**(0)⊥

_{P}**N**(0), these vectors are in the plane perpendicular to

_{Q}**t**and they are defined by α(

*v*), Fig. 3b.

*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

*F*(

_{j}*u*), where

*j*= 1,2,..14, for example

*F*(

_{1}*u*)≡(

**v**

_{i}-

**v**

_{x})

**·P**′(

*u*),

*F*(

_{2}*u*)≡(

**v**

_{i}-

**v**

_{x})

**·**(

**P**′(

*u*) ×

**N**(

_{P}*u*)), ...

*F*(

_{14}*u*)≡│

**N**(

_{Q}*u*)│-1. Hence we have

*F*(

_{j}*u*) = 0, for each

*j*. When the functions

**P**(

*u*),

**Q**(

*u*),

**N**(

_{P}*u*),

**N**(

_{Q}*u*),

*u*′(

*u*) and

*u*′′(

*u*) are analytic about

*u*= 0, then the functions

*F*(

_{j}*u*) are analytic as well, so that by Taylor’s theorem:

*j*= 1,2,..14 and

*i*= 0,1,...,

*n*.

*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**,

**N**,

_{P}**N**,

_{Q}*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

## References and links

1. | R. Winston, J. C. Miñano, and P. Benítez, |

2. | D. Grabovickić, P. Benítez, and J. C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express |

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 |

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. |

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 |

9. | B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics |

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

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

**ToC Category:**

Optical Design and Fabrication

**History**

Original Manuscript: March 2, 2011

Revised Manuscript: May 13, 2011

Manuscript Accepted: May 14, 2011

Published: June 3, 2011

**Citation**

Dejan Grabovičkić, Pablo Benítez, and Juan C. Miñano, "Free-form V-groove reflector design with the SMS method in three dimensions," Opt. Express **19**, A747-A756 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-S4-A747

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### References

- R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, (Elsevier Academic Press, 2004)
- 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]
- 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]
- International Patent Pending, US2010/002320 A1
- T. Levi-Civita, “Complementi al teorema di Malus-Dupin. Nota I,” Atti Accad. Sci. Torino, Cl. Sci. Fis, Mat. Nat. 9, 185–189 (1900).
- 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)
- A. Rabl, “Prisms with total internal reflection as solar reflectors,” Sol. Energy 19(5), 555–565 (1977). [CrossRef]
- B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).
- Synopsys software package LightTools, http://www.opticalres.com/

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