Abstract

A new optical component of transmissive faceted structure that allows the reshaping of collimated white light beam is depicted and realized. After being illuminated, each facet element in the structure can slightly tilt along its own axes to deflect the incident light. The calculation of the tilt angles is made by a numerical approach, which is the inverse solution of the analytical relationships between the two-dimensional tilt angles of the facet element and the local coordinates of the outgoing ray on the target plan. For a predefined illumination pattern on a fixed screen, the numerical computation of the 2D tilt angle matrix of the faceted structure is described. In the meantime, a Monte Carlo ray tracing program is created to calculate and verify the irradiance map, all compared with well-recognized commercial illumination software programs. Afterwards, the component is fabricated using an innovative additive technology, inspired by 3D printing and proposed by Luximprint.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  25. N. Bokor and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36(4), 1014–1020 (1997).
    [Crossref]
  26. N. Bokor and Z. Papp, “Computational method for testing computer-generated holograms,” Opt. Eng. 35(10), 2810–2815 (1996).
    [Crossref]
  27. B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
    [Crossref]

2015 (2)

2011 (2)

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

D. Michaelis, P. Schreiber, and A. Bräuer, “Cartesian oval representation of freeform optics in illumination systems,” Opt. Lett. 36(6), 918–920 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

J. C. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

2008 (5)

2007 (1)

C. H. Chen, C. C. Chen, and W. C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007).
[Crossref]

2006 (1)

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

2005 (2)

G. Patow and X. Pueyo, “A survey of inverse surface design from light transport behavior specification,” Comput. Graph. Forum 24(4), 773–789 (2005).
[Crossref]

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

2004 (1)

J. F. Van Derlofske and T. A. Hough, “Analytical model of flux propagation in light-pipe systems,” Opt. Eng. 43(7), 1503–1510 (2004).
[Crossref]

2002 (3)

N. Bokor and N. Davidson, “Anamorphic, adiabatic beam shaping of diffuse light using a tapered reflective tube,” Opt. Commun. 201(4–6), 243–249 (2002).
[Crossref]

J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 52–56 (2002).

H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
[Crossref] [PubMed]

1999 (1)

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

1998 (1)

S. R. David, C. T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” Proc. SPIE 3482, 437–446 (1998).
[Crossref]

1997 (1)

N. Bokor and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36(4), 1014–1020 (1997).
[Crossref]

1996 (1)

N. Bokor and Z. Papp, “Computational method for testing computer-generated holograms,” Opt. Eng. 35(10), 2810–2815 (1996).
[Crossref]

1989 (1)

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

1979 (1)

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Energy 22(2), 119–123 (1979).
[Crossref]

Akisawa, A.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

Allebach, J. P.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Asoubar, D.

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Benitez, P.

J. C. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Benítez, P.

Blen, J.

Bokor, N.

N. Bokor and N. Davidson, “Anamorphic, adiabatic beam shaping of diffuse light using a tapered reflective tube,” Opt. Commun. 201(4–6), 243–249 (2002).
[Crossref]

N. Bokor and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36(4), 1014–1020 (1997).
[Crossref]

N. Bokor and Z. Papp, “Computational method for testing computer-generated holograms,” Opt. Eng. 35(10), 2810–2815 (1996).
[Crossref]

Bräuer, A.

Cassarly, W. J.

F. Fournier, W. J. Cassarly, and J. P. Rolland, “Method to improve spatial uniformity with lightpipes,” Opt. Lett. 33(11), 1165–1167 (2008).
[Crossref] [PubMed]

S. R. David, C. T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” Proc. SPIE 3482, 437–446 (1998).
[Crossref]

Chen, C. C.

C. H. Chen, C. C. Chen, and W. C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007).
[Crossref]

Chen, C. H.

C. H. Chen, C. C. Chen, and W. C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007).
[Crossref]

Cheng, C. M.

Chern, J. L.

Crailsheim, H.

Dai, Y. J.

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

David, S. R.

S. R. David, C. T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” Proc. SPIE 3482, 437–446 (1998).
[Crossref]

Davidson, N.

N. Bokor and N. Davidson, “Anamorphic, adiabatic beam shaping of diffuse light using a tapered reflective tube,” Opt. Commun. 201(4–6), 243–249 (2002).
[Crossref]

Flury, M.

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Fontaine, J.

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Fournier, F.

Friesem, A. A.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Energy 22(2), 119–123 (1979).
[Crossref]

Gérard, P.

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Harnisch, E.

Hellmann, C.

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Hicks, R. A.

Hough, T. A.

J. F. Van Derlofske and T. A. Hough, “Analytical model of flux propagation in light-pipe systems,” Opt. Eng. 43(7), 1503–1510 (2004).
[Crossref]

Jennison, B. K.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Kashiwagi, T.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

Kim, J.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

Klein, J.

König, N.

Kritchman, E. M.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Energy 22(2), 119–123 (1979).
[Crossref]

Kuhn, M.

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Leutz, R.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

Liang, W. C.

C. H. Chen, C. C. Chen, and W. C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007).
[Crossref]

Michaelis, D.

Miñano, J. C.

Moreno, I.

Muschaweck, J.

O’Gallagher, J. J.

J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 52–56 (2002).

Papp, Z.

N. Bokor and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36(4), 1014–1020 (1997).
[Crossref]

N. Bokor and Z. Papp, “Computational method for testing computer-generated holograms,” Opt. Eng. 35(10), 2810–2815 (1996).
[Crossref]

Park, K. M.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

Patow, G.

G. Patow and X. Pueyo, “A survey of inverse surface design from light transport behavior specification,” Comput. Graph. Forum 24(4), 773–789 (2005).
[Crossref]

Pueyo, X.

G. Patow and X. Pueyo, “A survey of inverse surface design from light transport behavior specification,” Comput. Graph. Forum 24(4), 773–789 (2005).
[Crossref]

Rhee, J. G.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

Ries, H.

Rolland, J.

Rolland, J. P.

Russew, M.

Ryu, K.

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

Santamaria, A.

J. C. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Santamaría, A.

Schmitt, R.

Schreiber, P.

Schweitzer, H.

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Sumathy, K.

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Suzuki, A.

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

Takakura, Y.

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Twardworski, P.

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Van Derlofske, J. F.

J. F. Van Derlofske and T. A. Hough, “Analytical model of flux propagation in light-pipe systems,” Opt. Eng. 43(7), 1503–1510 (2004).
[Crossref]

Walker, C. T.

S. R. David, C. T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” Proc. SPIE 3482, 437–446 (1998).
[Crossref]

Wang, R. Z.

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Winston, R.

J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 52–56 (2002).

Wyrowski, F.

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Xie, W. T.

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Yekutieli, G.

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Energy 22(2), 119–123 (1979).
[Crossref]

Appl. Opt. (2)

Comput. Graph. Forum (1)

G. Patow and X. Pueyo, “A survey of inverse surface design from light transport behavior specification,” Comput. Graph. Forum 24(4), 773–789 (2005).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

N. Bokor and N. Davidson, “Anamorphic, adiabatic beam shaping of diffuse light using a tapered reflective tube,” Opt. Commun. 201(4–6), 243–249 (2002).
[Crossref]

M. Flury, P. Gérard, Y. Takakura, P. Twardworski, and J. Fontaine, “Investigation of M2 factor influence for paraxial computer-generated hologram reconstruction using a statistical method,” Opt. Commun. 248(4–6), 347–357 (2005).
[Crossref]

Opt. Eng. (4)

N. Bokor and Z. Papp, “Monte Carlo method in computer holography,” Opt. Eng. 36(4), 1014–1020 (1997).
[Crossref]

N. Bokor and Z. Papp, “Computational method for testing computer-generated holograms,” Opt. Eng. 35(10), 2810–2815 (1996).
[Crossref]

B. K. Jennison, J. P. Allebach, and D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28(6), 286629 (1989).
[Crossref]

J. F. Van Derlofske and T. A. Hough, “Analytical model of flux propagation in light-pipe systems,” Opt. Eng. 43(7), 1503–1510 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Opt. Mater. Express (1)

Opt. Rev. (2)

C. H. Chen, C. C. Chen, and W. C. Liang, “Light pipe line beam shaper,” Opt. Rev. 14(4), 231–235 (2007).
[Crossref]

J. C. Miñano, P. Benitez, and A. Santamaria, “Free-form optics for illumination,” Opt. Rev. 16(2), 99–102 (2009).
[Crossref]

Proc. SPIE (3)

S. R. David, C. T. Walker, and W. J. Cassarly, “Faceted reflector design for uniform illumination,” Proc. SPIE 3482, 437–446 (1998).
[Crossref]

J. J. O’Gallagher and R. Winston, “Nonimaging solar concentrator with near uniform irradiance for photovoltaic arrays,” Proc. SPIE 4446, 52–56 (2002).

D. Asoubar, C. Hellmann, H. Schweitzer, M. Kuhn, and F. Wyrowski, “Customized homogenization and shaping of LED light by micro cells arrays,” Proc. SPIE 9383, 93831B (2015).
[Crossref]

Renew. Sustain. Energy Rev. (1)

W. T. Xie, Y. J. Dai, R. Z. Wang, and K. Sumathy, “Concentrated solar energy applications using Fresnel lenses: A review,” Renew. Sustain. Energy Rev. 15(6), 2588–2606 (2011).
[Crossref]

Sol. Energy (3)

K. Ryu, J. G. Rhee, K. M. Park, and J. Kim, “Concept and design of modular Fresnel lenses for concentration solar PV system,” Sol. Energy 80(12), 1580–1587 (2006).
[Crossref]

E. M. Kritchman, A. A. Friesem, and G. Yekutieli, “Efficient Fresnel lens for solar concentration,” Sol. Energy 22(2), 119–123 (1979).
[Crossref]

R. Leutz, A. Suzuki, A. Akisawa, and T. Kashiwagi, “Design of a nonimaging Fresnel lens for solar concentrators,” Sol. Energy 65(6), 379–387 (1999).
[Crossref]

Other (2)

T. Weyrich, P. Peers, W. Matusik, and S. Rusinkiewicz, “Fabricating microgeometry for custom surface reflectance,” ACM Transactions on Graphics (Proc. SIGGRAPH) 28, 321–326 (2009).
[Crossref]

Luximprint, Additive Fabrication Services for Functional and Decorative Optical Plastics, http://www.luximprint.com .

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Figures (13)

Fig. 1
Fig. 1 General 2D view of the optical configuration with illumination source (S), transmissive faceted structure (FS), and the detector (D).
Fig. 2
Fig. 2 2D view in the y-z plan and the x-z plan, when Fm,n deflects with (αm,n, βm,n) along the axis x and axis y. (a) The position of F(m, n, 0, v) shown in y-z plan. (b) The position of F (m, n, u, 0) in x-z plan.
Fig. 3
Fig. 3 Steps for local illumination in Matlab and global illumination in Zemax and LightTools.
Fig. 4
Fig. 4 Desired illumination pattern S4 on a plane.
Fig. 5
Fig. 5 The raytracing configuration in OpticStudio 16 and the 3D scheme of the faceted structure.
Fig. 6
Fig. 6 The geometrical information of the faceted structure.
Fig. 7
Fig. 7 Normalized spectral intensity of Ocean Optics 2000 in [0.26, 1] μm.
Fig. 8
Fig. 8 Illumination pattern S4. (a) Local illumination result in Matlab. (b) Global illumination result in Zemax. (c) Global illumination result in LightTools.
Fig. 9
Fig. 9 3D and 1D irradiance of illumination pattern S4 in LightTools.
Fig. 10
Fig. 10 2D irradiance of illumination patterns in LightTools. (a) Irradiance of four symmetric squares with new arrangement. (b) Irradiance of rectangular hole with 20 mm inner hole size.
Fig. 11
Fig. 11 Spectral sensitivity of transmissive faceted structure to Ocean Optics HL 2000.
Fig. 12
Fig. 12 2D diagram of the optical system in the experiment.
Fig. 13
Fig. 13 Experiment results. (a) Irradiance map taken by the camera before the tracing paper. (b) Irradiance map after the tracing paper.

Tables (2)

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Table 1 Design Parameters

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Table 2 Quality Evaluation of Illumination Pattern S4

Equations (18)

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F ( m,n,u,v )=( ( N Fx 1 ) p x 2 +( m1 ) p x + u p x cos β m,n 2 ( N Fy 1 ) p y 2 +( n1 ) p y + v p y cos α m,n 2 u p x sin β m,n 2 + v p y sin α m,n 2 ).
n F = (tan β m,n ,tan α m,n ,1) 1+ tan 2 β m,n + tan 2 α m,n ,
t = n 1 n 2 s +( n 1 n 2 cos θ i 1 sin 2 θ t ) n F ,
s =( 0 0 1 ),
cos θ i = n F s = 1 1+ tan 2 β m,n + tan 2 α m,n .
sin 2 θ t = ( n 1 ) 2 ( 1 1 1+ tan 2 β m,n + tan 2 α m,n ).
t = n 1 s +( n 1 cos θ i 1 sin 2 θ t ) n F =( btan β m,n btan α m,n n 1 b ),
b= n 1 1+ tan 2 β m,n + tan 2 α m,n n 1 2 ( tan 2 β m,n + tan 2 α m,n ) 1+ tan 2 β m,n + tan 2 α m,n .
{ x= F ( m,n,u,v ) x + t x k y= F ( m,n,u,v ) y + t y k z= F ( m,n,u,v ) z + t z k ,
{ l D /2 x l D /2 l D /2 y l D /2 z= d FD .
k= d FD F ( m,n,u,v ) z t z ,
{ D ( m,n,u,v ) X = f x ( α m,n , β m,n )= F ( m,n,u,v ) x + t x k D ( m,n,u,v ) Y = f y ( α m,n , β m,n )= F ( m,n,u,v ) y + t y k .
{ α m,n = f α 1 ( X D (m,n), Y D (m,n) ) β m,n = f β 1 ( X D (m,n), Y D (m,n) ) .
{ x F [ 1, N ray ]=( N Fx p x )rand[ 1, N ray ] N Fx p x /2 y F [ 1, N ray ]=( N Fy p y )rand[ 1, N ray ] N Fy p y /2 ,
{ m[ 1, N ray ]=ceil[ N Fx rand[ 1, N ray ] ] n[ 1, N ray ]=ceil[ N Fy rand[ 1, N ray ] ] ,
{ u[ 1, N ray ]= 2 p x ( x F [ 1, N ray ]m[ 1, N ray ] p x +( N Fx +1 ) p x 2 ) v[ 1, N ray ]= 2 p y ( y F [ 1, N ray ]n[ 1, N ray ] p y +( N Fy +1 ) p y 2 ) ,
{ X p [ 1, N ray ]= N D /2 +round( X D [ 1, N ray ] l D / N D ) Y p [ 1, N ray ]= N D /2 +round( X D [ 1, N ray ] l D / N D ) .
C= c 1 c 2 c 3 c 4 , where c 1 = pD I M (p) I Z/L (p) N D N D ,  c 2 =[ pD I M (p) N D N D ] pD I Z/L (p) N D N D , c 3 = pD I M 2 (p) N D N D [ pD I M (p) N D N D ] 2 , c 4 = pD I Z/L 2 (p) N D N D [ pD I Z/L (p) N D N D ] 2 .

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