Abstract

Presented is a quasi-analytic method for irradiance evaluation through a single refractive surface from a single Lambertian source. The method is compared to Monte-Carlo raytracing for a sample system, producing in much less time an irradiance distribution equal to within the latter’s statistical noise. In addition to its interest to optical analysis, the method is also useful for optical design problems by allowing for fast, noise-free evaluation of merit functions, along with their derivatives.

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

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References

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  1. R. Wester and A. Bäuerle, “Light Shaping for Illumination,” Adv. Opt. Technol. 2, 301–311 (2013).
  2. W. J. Cassarly, “Illumination merit functions,” Proc. SPIE 6670, 66700K (2007).
    [Crossref]
  3. E. M. Sparrow, “A New and Simpler Formulation for Radiative Angle Factors,” J. Heat Transf. 85(2), 81–87 (1963).
    [Crossref]
  4. D. G. Koch, “Simplified irradiance/illuminance calculations in optical systems,” Proc. SPIE 1780, 17800F (1992).
  5. H. Ries and J. Muschaweck, “Tailored freeform optical surfaces,” J. Opt. Soc. Am. A 19(3), 590–595 (2002).
    [Crossref]
  6. Steven G. Johnson, “The NLopt nonlinear-optimization package,” http://ab-initio.mit.edu/nlopt .
  7. H. C. Hottel and A. F. Sarofim, Radiative Transfer, (1969, McGraw-Hill).
  8. Synopsys Inc., “LightTools® 8.5,” https://optics.synopsys.com/lighttools .
  9. R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
    [Crossref]
  10. A. Hirst and J. Muschaweck, “Irradiance tailoring for extended sources in 3D by implicit integral equation solution,” OSA Tech. Dig. (online), FT4B.2 (2015).

2014 (1)

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

2013 (1)

R. Wester and A. Bäuerle, “Light Shaping for Illumination,” Adv. Opt. Technol. 2, 301–311 (2013).

2007 (1)

W. J. Cassarly, “Illumination merit functions,” Proc. SPIE 6670, 66700K (2007).
[Crossref]

2002 (1)

1992 (1)

D. G. Koch, “Simplified irradiance/illuminance calculations in optical systems,” Proc. SPIE 1780, 17800F (1992).

1963 (1)

E. M. Sparrow, “A New and Simpler Formulation for Radiative Angle Factors,” J. Heat Transf. 85(2), 81–87 (1963).
[Crossref]

Bäuerle, A.

R. Wester and A. Bäuerle, “Light Shaping for Illumination,” Adv. Opt. Technol. 2, 301–311 (2013).

Berens, M.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Cassarly, W. J.

W. J. Cassarly, “Illumination merit functions,” Proc. SPIE 6670, 66700K (2007).
[Crossref]

Hirst, A.

A. Hirst and J. Muschaweck, “Irradiance tailoring for extended sources in 3D by implicit integral equation solution,” OSA Tech. Dig. (online), FT4B.2 (2015).

Hottel, H. C.

H. C. Hottel and A. F. Sarofim, Radiative Transfer, (1969, McGraw-Hill).

Koch, D. G.

D. G. Koch, “Simplified irradiance/illuminance calculations in optical systems,” Proc. SPIE 1780, 17800F (1992).

Loosen, P.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Müller, G.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Muschaweck, J.

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

A. Hirst and J. Muschaweck, “Irradiance tailoring for extended sources in 3D by implicit integral equation solution,” OSA Tech. Dig. (online), FT4B.2 (2015).

Ries, H.

Sarofim, A. F.

H. C. Hottel and A. F. Sarofim, Radiative Transfer, (1969, McGraw-Hill).

Sparrow, E. M.

E. M. Sparrow, “A New and Simpler Formulation for Radiative Angle Factors,” J. Heat Transf. 85(2), 81–87 (1963).
[Crossref]

Stollenwerk, J.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Völl, A.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Wester, R.

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

R. Wester and A. Bäuerle, “Light Shaping for Illumination,” Adv. Opt. Technol. 2, 301–311 (2013).

Adv. Opt. Technol. (1)

R. Wester and A. Bäuerle, “Light Shaping for Illumination,” Adv. Opt. Technol. 2, 301–311 (2013).

J. Heat Transf. (1)

E. M. Sparrow, “A New and Simpler Formulation for Radiative Angle Factors,” J. Heat Transf. 85(2), 81–87 (1963).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (1)

R. Wester, G. Müller, A. Völl, M. Berens, J. Stollenwerk, and P. Loosen, “Designing optical free-form surfaces for extended sources,” Opt. Express 22, 552–560 (2014).
[Crossref]

Proc. SPIE (2)

D. G. Koch, “Simplified irradiance/illuminance calculations in optical systems,” Proc. SPIE 1780, 17800F (1992).

W. J. Cassarly, “Illumination merit functions,” Proc. SPIE 6670, 66700K (2007).
[Crossref]

Other (4)

Steven G. Johnson, “The NLopt nonlinear-optimization package,” http://ab-initio.mit.edu/nlopt .

H. C. Hottel and A. F. Sarofim, Radiative Transfer, (1969, McGraw-Hill).

Synopsys Inc., “LightTools® 8.5,” https://optics.synopsys.com/lighttools .

A. Hirst and J. Muschaweck, “Irradiance tailoring for extended sources in 3D by implicit integral equation solution,” OSA Tech. Dig. (online), FT4B.2 (2015).

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Figures (3)

Fig. 1
Fig. 1 A planar Lambertian source is immersed in an initial optical medium, separated by a freeform surface from a second medium, wherein a target point lies whose irradiance value is to be determined. The visible angular region (or beam-print) of the source on the surface, as seen by this target point, is projected into the target surface’s local space.
Fig. 2
Fig. 2 On the left is a wireframe render of the test profile used for verification. Of particular significance is the saddle point around the centre. On the right is a top-down view of the surface, showing a subset of solved beam-prints between the test source and target, whose sizes and shapes are determined by their relative orientation to and size of the target surface.
Fig. 3
Fig. 3 Comparison plot of the target plane irradiance through the test surface. On the left, unsmoothed results from Monte-Carlo raytracing, which are very noisy even at 2.5 × 107 rays. On the right, unsmoothed results from the beam-print method at N = 20, which are simultaneously in agreement with and less noisy than those from Monte-Carlo raytracing.

Tables (1)

Tables Icon

Table 1 The execution time increases linearly with N. Results at N = 200 are considered to have converged to the quasi-analytic result. Comparison to LightTools is underneath, along with LightTools’ own estimate for the average Monte-Carlo statistical noise.

Equations (7)

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E = L n t 2 Ω = L n t 2 Ω [ R ^ N t ] d Ω ,
Ω = 1 2 C [ N t × ( t s ) t s 2 d s c ] 1 2 [ N t C ( t s ) × d s c t s 2 ] ,
u c , v c = arg min u , v [ 0 , 1 ] { OPL ( r c , s ( u , v ) , t ) } , where OPL ( r , s , t ) = n r s r + n t t s .
Ω = 1 2 [ N t i = 1 N Γ i ] , where Γ i Angle ( R i , R i 1 ) ( R i × R i 1 ) R i × R i 1 ,
u i , v i = arg min u , v [ 0 , 1 ] { OPL ( r i , s ( u , v ) , t ) } ,
| Δ f n | | f n | < ϵ f , Δ x n x n < ϵ x ,
% Erro = ( 100 % × ) | E N E N = 200 | | E N = 200 | ,

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