## Aspheric V-groove reflector design with the SMS method in two dimensions

Optics Express, Vol. 18, Issue 3, pp. 2515-2521 (2010)

http://dx.doi.org/10.1364/OE.18.002515

Acrobat PDF (597 KB)

### Abstract

The Simultaneous Multiple Surface design method in two dimensions (SMS2D) is applied to the design of aspheric V-groove reflectors. The general design problem is to achieve perfect coupling of two wavefronts after two reflections at the groove, no matter which side of the groove the rays hit first. Two types of configurations are identified, and several symmetric and asymmetric design examples are given. Computer simulations with a commercial simulation package are also shown.

© 2010 OSA

## 1. Introduction

*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.

*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

2. Y. Wang, K. Li, and S. Inatsugu, “New retroreflector technology for light-collecting systems,” Opt. Eng. **46**(8), 084001 (2007). [CrossRef]

*i.e.*admit Taylor series expansion). These solutions [3] are calculated by the Simultaneous Multiple Design method in two dimensions (SMS2D).

## 2. Statement of the problem

*y*) but, in general, the profiles are not symmetric.

4. B. van-Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A **11**(11), 2905–2914 (1994). [CrossRef]

*h*) and that two centers of the wavefronts are points (

*x*, 0) and (-

_{A}*x*, 0), where

_{B}*x*,

_{A}*x*and

_{B}*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*, 0) and (

_{A}*g*(

*ε*),

*ε*) are fixed, then Fermat’s principle implies thatwhere

*f*(

*y*),

*y*) and (

*x*, 0) must satisfywhere

_{A}*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 where

*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:

^{¢}(0) = tan

*α*and g

^{¢}(0) = −tan

*β*, where

*α*+

*β*= π/2.

*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].

*WF*and

_{A}*WF*can be done by substituting functions

_{B}*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

### 3.1 Polynomial approximation near the groove peak

*F*= 0, where

_{j}*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:

*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*.

*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

_{0}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 P

_{0}. Once {P

_{0}, n

_{P0}} are prescribed we are able to calculate a point on the other surface Q

_{0}. The point Q

_{0}is calculated along the trajectory of the ray from A, after the reflection at P

_{0}being the one satisfying that the total optical path length equals

_{Q0}is then calculated which produces the reflection from P

_{0}-Q

_{0}to Q

_{0}-A. The procedure continues thereafter using the ray form B impinging at the point {Q

_{0}, n

_{Q0}}, thereby calculating the next point P

_{1}of the first surface (Fig. 4 the ray in blue), using the optical path length

_{1}, P

_{2}, Q

_{2},

*etc.*along the curves (with the y coordinate of the points decreasing thereby).

_{-1}, Q

_{-1}, P

_{-2},

*etc.*are obtained when one starts the SMS procedure using the point P

_{0}and the ray from B (instead of the ray from A, as before). The sequence of points {…, P

_{-2}, Q

_{-1}, P

_{-1}, Q

_{0}, P

_{0}, Q

_{1}, P

_{2}, Q

_{2}, …} together with their associated normal vectors is called an SMS chain, in the SMS nomenclature. The sequence {P

*} results converge to the groove peak for i→-∞. The sequence {P*

_{i}*} for i→ + ∞ does not, however, generally converge.*

_{i}_{0}, n

_{P0}} we choose two consecutive points of the sequence on one side, for example, P

_{0}and P

_{1}. Interpolating a C

^{1}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 P

_{0}close enough to the groove peak.

_{0}P

_{1}with the same optical path length, a new set of points on the other surface is calculated, between points Q

_{0}and Q

_{1}(Fig. 4, those rays indicated by red dashed lines). Repeating the same process, now using rays from B and the points of segment Q

_{0}Q

_{1}, we obtain the points of the segment P

_{1}P

_{2}. This procedure can be implemented until the all segments are filled by new points. The number of the points of the first segment P

_{0}P

_{1}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

*α*= π/4 and the asymmetric solution to

*α*≠0.

### 4.2 Circular caustic

7. R. Winston, “Cavity Enhancement by controlled directional Scattering,” Appl. Opt. **19**(2), 195–197 (1980). [CrossRef] [PubMed]

9. H. Ries and J. Muschaweck, “Double-tailored microstructures,” SPIE **3781**, 124–128 (1999). [CrossRef]

*WF*and

_{A}*WF*define a common circular caustic. Then, the functions (

_{B}*x*(

_{A}*t*),

*y*(

_{A}*t*)) and (

*x*(

_{B}*t’*),

*y*(

_{B}*t’*)) are two different parameterizations of the circle.

## 5. Summary

## Acknowledgments

## References and links

1. | R. Leutz, L. Fu, and H. Ries, “Carambola reflector for recycling the light,” Appl. Opt. |

2. | Y. Wang, K. Li, and S. Inatsugu, “New retroreflector technology for light-collecting systems,” Opt. Eng. |

3. | Patent pending. |

4. | B. van-Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A |

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

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

7. | R. Winston, “Cavity Enhancement by controlled directional Scattering,” Appl. Opt. |

8. | W. R. McIntire, “Elimination of the optical Losses Due to Gaps Between Absorbers and Their Reflectors”, Proc. 1980 Ann.Meeeting 3.1:600. AS Int. Solar Energy Society (1980) |

9. | H. Ries and J. Muschaweck, “Double-tailored microstructures,” SPIE |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

**History**

Original Manuscript: October 22, 2009

Revised Manuscript: January 13, 2010

Manuscript Accepted: January 17, 2010

Published: January 22, 2010

**Virtual Issues**

Focus Issue: Solar Concentrators (2010) *Optics Express*

**Citation**

Dejan Grabovičkić, Pablo Benítez, and Juan C. Miñano, "Aspheric V-groove reflector design with the SMS method in two dimensions," Opt. Express **18**, 2515-2521 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-3-2515

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### References

- R. Leutz, L. Fu, and H. Ries, “Carambola reflector for recycling the light,” Appl. Opt. 45, 572–2575 (2005).
- Y. Wang, K. Li, and S. Inatsugu, “New retroreflector technology for light-collecting systems,” Opt. Eng. 46(8), 084001 (2007). [CrossRef]
- Patent pending.
- B. van-Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A 11(11), 2905–2914 (1994). [CrossRef]
- B. van-Brunt, “An existence, uniqueness and analyticity theorem for a class of functional differential equations,” NZ J. Mathematics 22, 101–107 (1993).
- R. Winston, J. C. Miñano, and P. Benítez, Nonimaging Optics, pp. 186–187, (Elsevier, Academic, Press 2004)
- R. Winston, “Cavity Enhancement by controlled directional Scattering,” Appl. Opt. 19(2), 195–197 (1980). [CrossRef] [PubMed]
- W. R. McIntire, “Elimination of the optical Losses Due to Gaps Between Absorbers and Their Reflectors”, Proc. 1980 Ann.Meeeting 3.1:600. AS Int. Solar Energy Society (1980)
- H. Ries and J. Muschaweck, “Double-tailored microstructures,” SPIE 3781, 124–128 (1999). [CrossRef]

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