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.

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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/

2010

1999

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

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

1977

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

1900

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.

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]

Grabovickic, D.

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.

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]

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.

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

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

Opt. Express

Proc. SPIE

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

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

Other

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

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

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)

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Figures (8)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Free-form V-groove reflector.

Fig. 3
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
Fig. 4

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

Fig. 5
Fig. 5

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

Fig. 6
Fig. 6

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

Fig. 7
Fig. 7

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

Fig. 8
Fig. 8

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

Equations (16)

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( 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.

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