Abstract

We obtain simple exact formulas for both caustic and refracted wavefronts through smooth surfaces by considering an incident plane wavefront propagating along the optical axis, providing a condition for total internal reflection (TIR). On the other hand, a formula to provide the maximum slopes for refracted rays outside of the lens is directly related to the condition to obtain the inflection points on the refracting surface. Demanding simultaneously both conditions can potentially provide the shape for an optimized surface which reduces the gap produced by TIR and to refract efficiently all the light outside of the lens. This has a wide potential in applications on the field of non-imagining systems and illumination.

© 2015 Optical Society of America

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References

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  1. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its RamificationsWiley-VCH Verlag GmbH & Co. KGaA, 2006), pp. 179–186.
    [Crossref]
  2. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper, Proc. SPIE 6668, 666805 (2007).
    [Crossref]
  3. D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
    [Crossref]
  4. J. Stoker, Differential Geometry (Wiley-Interscience, 1969), pp. 12–52.
  5. M. Avendaño-Alejo, “Caustics in a meridional plane produced by plano-convex aspheric lenses,” J. Opt. Soc. Am. A 30, 501–508 (2013).
    [Crossref]
  6. M. Avendaño-Alejo, “Caustic surface of a plane wavefront refracted by a smooth arbitrary surface,” Proc. SPIE 9192, 919208 (2014).
    [Crossref]
  7. E. W. Weisstein, “Cycloid Catacaustic.” http://mathworld.wolfram.com/CycloidCatacaustic.html
  8. A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 2000), pp. 37–53.
  9. M. Avendaño-Alejo, L. Castañeda, and N. Qureshi, “Huygens’ Principle: Exact wavefronts produced by aspheric lenses,” Opt. Express 21, 29874–29884 (2013).
    [Crossref]
  10. S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), pp. 11–35.
  11. B. Kim, J. Kim, O. Won-Suk, and S. Kang, “Eliminating hotspots in a multi-chip LED array direct backlight system with optimal patterned reflectors for uniform illuminance and minimal system thickness,” Opt. Express 18, 8595–8604 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. C. Zuo, J. Sun, J. Zhang, Y. Hu, and Q. Chen, “Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix,” Opt. Express 23, 14314–14328 (2015).
    [Crossref] [PubMed]
  14. M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
    [Crossref]
  15. D. Korsch, Reflective Optics (Academic Press, Inc., 1991) pp. 261–268.
    [Crossref]

2015 (1)

2014 (1)

M. Avendaño-Alejo, “Caustic surface of a plane wavefront refracted by a smooth arbitrary surface,” Proc. SPIE 9192, 919208 (2014).
[Crossref]

2013 (2)

2010 (2)

2008 (1)

2007 (1)

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper, Proc. SPIE 6668, 666805 (2007).
[Crossref]

2000 (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[Crossref]

Avendaño-Alejo, M.

Castañeda, L.

Chen, Q.

Cornbleet, S.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), pp. 11–35.

Feng, Z.

Gray, A.

A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 2000), pp. 37–53.

Han, Y.

Hoffnagle, J. A.

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[Crossref]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper, Proc. SPIE 6668, 666805 (2007).
[Crossref]

Hu, Y.

Juskaitis, R.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[Crossref]

Kang, S.

Kim, B.

Kim, J.

Korsch, D.

D. Korsch, Reflective Optics (Academic Press, Inc., 1991) pp. 261–268.
[Crossref]

Li, H.

Luo, Y.

Neil, M. A. A.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[Crossref]

Qureshi, N.

Shealy, D. L.

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustics of a plane wave refracted by an arbitrary surface,” J. Opt. Soc. Am. A 25, 2370–2382 (2008).
[Crossref]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper, Proc. SPIE 6668, 666805 (2007).
[Crossref]

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its RamificationsWiley-VCH Verlag GmbH & Co. KGaA, 2006), pp. 179–186.
[Crossref]

Stoker, J.

J. Stoker, Differential Geometry (Wiley-Interscience, 1969), pp. 12–52.

Sun, J.

Wilson, T.

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[Crossref]

Won-Suk, O.

Zhang, J.

Zuo, C.

J. Microscopy (1)

M. A. A. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy 197, 219–223 (2000).
[Crossref]

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

Opt. Express (4)

Proc. SPIE (2)

M. Avendaño-Alejo, “Caustic surface of a plane wavefront refracted by a smooth arbitrary surface,” Proc. SPIE 9192, 919208 (2014).
[Crossref]

D. L. Shealy and J. A. Hoffnagle, “Wavefront and caustic surfaces of refractive laser beam shaper, Proc. SPIE 6668, 666805 (2007).
[Crossref]

Other (6)

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics, The K-function and its RamificationsWiley-VCH Verlag GmbH & Co. KGaA, 2006), pp. 179–186.
[Crossref]

J. Stoker, Differential Geometry (Wiley-Interscience, 1969), pp. 12–52.

E. W. Weisstein, “Cycloid Catacaustic.” http://mathworld.wolfram.com/CycloidCatacaustic.html

A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 2000), pp. 37–53.

S. Cornbleet, Microwave and Optical Ray Geometry (Wiley, 1984), pp. 11–35.

D. Korsch, Reflective Optics (Academic Press, Inc., 1991) pp. 261–268.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 (a) A smooth surface and their parameters involved in the process of refraction. (b). Exact ray tracing through a smooth lens and their associated caustics.
Fig. 2
Fig. 2 (a) Different curate cycloids for τ ∈ [−π, 3π] with arbitrary dimensions. (b) A smooth lens considering a cycloid with a = 5.0, b = 0.88, for τ ∈ [−12π, 20π].
Fig. 3
Fig. 3 (a) a = 5cm, b = 0.94cm, with θmax = 48.189°. (b) a = 5cm, b = 0.88045cm, with θmax = 48.189°. (c) a = 5cm, b = 0.82cm, with θmax = 33.7731°, for τ ∈ [−π, 3π].
Fig. 4
Fig. 4 Behavior of the slopes for refracted rays outside of the smooth surfaces showing either continuity or discontinuity through the surface.
Fig. 5
Fig. 5 (a) Process of refraction produced by a smooth lens, and its associated parameters. (b) Wavelets and the archetype wavefront produced by a smooth surface, considering a plane wavefront incident on the lens.
Fig. 6
Fig. 6 Propagation of refracted wavefronts through two cycloids with the following parameters: (a) a = 5cm, b = 0.82cm. (b) a = 5cm, b = 0.88045cm, for τ ∈ [−π, 3π]

Equations (26)

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S = ( ( τ ) , G ( τ ) ) ,
y cos ( θ a θ i ) + z sin ( θ a θ i ) = G cos ( θ a θ i ) + [ t + ] sin ( θ i θ i ) ,
θ i = arctan [ τ G τ ] , θ a = arcsin [ n i τ n a τ 2 + G τ 2 ] ,
y sin ( θ a θ i ) + z cos ( θ a θ i ) = G sin ( θ a θ i ) + [ t + ] cos ( θ i θ i ) + ,
= G τ cos ( θ a θ i ) + τ sin ( θ a θ i ) θ a / τ θ i / τ .
z d ( τ ) = t + + [ n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 ] [ n a 2 G τ + n i n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 ] n a 2 ( n i 2 n a 2 ) [ τ G τ τ G τ τ τ ] , y d ( τ ) = G + [ n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 ] τ n a 2 [ τ G τ τ G τ τ τ ] ,
( z c ( τ ) , y c ( τ ) ( + G τ [ τ 2 G τ 2 ] 2 ( τ G τ τ τ τ G τ ) , G + τ G τ 2 τ G τ τ τ τ G τ ) ,
( z c ( r ) , y c ( r ) ) = ( z + 1 z 2 2 z , r z z ) ,
± n a n i 2 n a 2 = [ τ G τ ] .
( G , ) = ( b 2 π ( 1 + cos τ ) , a 10 π ( τ π b a sin τ ) ) , τ [ π , 3 π ] .
( τ , G τ ) = ( b sin τ 2 π , a b cos τ 10 π ) , ( τ τ , G τ τ ) = ( b cos τ 2 π , a b sin τ 10 π ) .
( z c c , y c c ) = ( t + + Δ [ n a 2 ( a b cos τ ) + n i Δ ] 50 π b n a 2 ( n i 2 n a 2 ) [ b a cos τ ] , G + Δ sin τ 10 π n a 2 [ b a cos τ ] ) ,
Δ = n a 2 ( a b cos τ ) 2 + 25 b 2 ( n a 2 n i 2 ) sin 2 τ .
d [ θ a θ i ] d τ = [ ( n i 2 n a 2 ) n i G τ + n a 2 G τ 2 + ( n i 2 n a 2 ) τ 2 ] [ G τ τ τ τ G τ τ n a 2 G τ 2 + ( n i 2 n a 2 ) τ 2 ] = 0 ,
G τ τ τ τ G τ τ = 0 , n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 = 0.
b = a cos τ .
τ τ G τ τ = ± n a n i 2 n a 2 ,
τ = ± arctan [ 5 n i 2 n a 2 n a ] ,
b = a [ n a 25 n i 2 n a 2 ] ,
( Z [ t + ] ) 2 + ( Y G ) 2 = ( n i n a ) 2 , G [ H , H ] .
Y G = { Z [ t + ] + ( n i n a ) 2 } [ τ G τ ] ,
Z ± ( τ ) = t + n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 n a 2 [ τ 2 + G τ 2 ] [ n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 ± n i G τ ] , Y ± ( τ ) = G n i τ n a 2 [ τ 2 + G τ 2 ] [ n i G τ ± n a 2 G τ 2 + ( n a 2 n i 2 ) τ 2 ] ,
Z w ( τ ) = t + ( n a 2 n i 2 ) n a 2 G τ 2 + [ n a 2 n i 2 ] τ 2 n a 2 [ n i G τ + n a 2 G τ 2 + [ n a 2 n i 2 ] τ 2 ] , Y w ( τ ) = G + n i ( n a 2 n i 2 ) τ n a 2 [ n i G τ + n a 2 G τ 2 + [ n a 2 n i 2 ] τ 2 ] ,
W = ( F ± [ G / τ ] ( F / τ ) 2 + ( G / τ ) 2 , G [ F / τ ] ( F / τ ) 2 + ( G / τ ) 2 ) .
Z w ( τ ) = Z w + [ n a 2 G τ + n i n a 2 G τ 2 + [ n a 2 n i 2 ] [ n i G τ + n a 2 G τ 2 + [ n a 2 n i 2 ] τ 2 ] ] n a , Y w ( τ ) = Y w [ ( n a 2 n i 2 ) τ [ n i G τ + n a 2 G τ 2 + [ n a 2 n i 2 ] τ 2 ] ] n a ,
Z w z ( τ ) = t + n a 3 + { 2 π n i n a ( n a 2 n i 2 ) [ 1 + cos τ ] b } Δ 2 π n a 2 [ n i [ a b cos τ ] + Δ ] , Z w z ( τ ) = a 10 π [ τ π b a sin τ ] 5 b ( n a 2 n i 2 ) { n i [ 1 + cos τ ] b + 2 π n a } sin τ 2 π n a 2 [ n i [ a b cos τ ] + Δ ] ,

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