Abstract

We present a method for designing lenses with two aspherical surfaces having minimal Fresnel losses among the class of stigmatic lenses. Minimization of Fresnel losses is achieved by ensuring equal ray deviation angles on the lens surfaces. Calculation of lenses with minimal Fresnel losses is reduced to solving an explicit ordinary differential equation. Simple analytical approximations are also obtained for the lens profiles.

© 2021 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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  15. M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23, A1140–A1148 (2015).
    [Crossref]
  16. S. V. Kravchenko, E. V. Byzov, M. A. Moiseev, and L. L. Doskolovich, “Design of optical elements with two refractive surfaces to generate a prescribed intensity distribution,” Comput. Opt. 39(4), 508–514 (2015).
    [Crossref]
  17. S. V. Kravchenko, M. A. Moiseev, E. V. Byzov, and L. L. Doskolovich, “Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions,” Opt. Commun. 459, 124976 (2020).
    [Crossref]
  18. L. L. Doskolovich, D. A. Bykov, K. V. Andreeva, and N. L. Kazanskiy, “Design of an axisymmetrical refractive optical element generating required illuminance distribution and wavefront,” J. Opt. Soc. Am. A 35, 1949–1953 (2018).
    [Crossref]
  19. M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29, 1758–1763 (2012).
    [Crossref]
  20. https://www.zemax.com/ .
  21. S. T. Bobrov, G. I. Greisukh, and Y. G. Tyrkevich, Optics of Diffractive Elements and Systems (Mashinostroenie, 1986).

2021 (1)

R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutierrez-Vega, “Exact equations for stigmatic singlet design meeting the Abbe sine condition,” Opt. Commun. 479, 126415 (2021).
[Crossref]

2020 (4)

2019 (1)

2018 (3)

2017 (1)

2016 (1)

2015 (4)

2012 (1)

1992 (1)

Achtner, B.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

Andreeva, K. V.

Artal, P.

Atchison, D. A.

Avendaño-Alejo, M.

Blechinger, F.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

Bobrov, S. T.

S. T. Bobrov, G. I. Greisukh, and Y. G. Tyrkevich, Optics of Diffractive Elements and Systems (Mashinostroenie, 1986).

Born, M.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1959).

Bykov, D. A.

Byzov, E. V.

S. V. Kravchenko, M. A. Moiseev, E. V. Byzov, and L. L. Doskolovich, “Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions,” Opt. Commun. 459, 124976 (2020).
[Crossref]

M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23, A1140–A1148 (2015).
[Crossref]

S. V. Kravchenko, E. V. Byzov, M. A. Moiseev, and L. L. Doskolovich, “Design of optical elements with two refractive surfaces to generate a prescribed intensity distribution,” Comput. Opt. 39(4), 508–514 (2015).
[Crossref]

Canioni, L.

Chaparro-Romo, H. A.

R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutierrez-Vega, “Exact equations for stigmatic singlet design meeting the Abbe sine condition,” Opt. Commun. 479, 126415 (2021).
[Crossref]

R. G. González-Acuña and H. A. Chaparro-Romo, “General formula for bi-aspheric singlet lens design free of spherical aberration,” Appl. Opt. 57, 9341–9345 (2018).
[Crossref]

Chassagne, B.

Doskolovich, L. L.

Fang, F.

Feuermann, D.

González-Acuña, R. G.

Gordon, J. M.

Gordon, M. J.

Greisukh, G. I.

S. T. Bobrov, G. I. Greisukh, and Y. G. Tyrkevich, Optics of Diffractive Elements and Systems (Mashinostroenie, 1986).

Gross, H.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

Gutierrez-Vega, J. C.

R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutierrez-Vega, “Exact equations for stigmatic singlet design meeting the Abbe sine condition,” Opt. Commun. 479, 126415 (2021).
[Crossref]

Gutiérrez-Vega, J. C.

Kazanskiy, N. L.

Kravchenko, S. V.

S. V. Kravchenko, M. A. Moiseev, E. V. Byzov, and L. L. Doskolovich, “Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions,” Opt. Commun. 459, 124976 (2020).
[Crossref]

S. V. Kravchenko, E. V. Byzov, M. A. Moiseev, and L. L. Doskolovich, “Design of optical elements with two refractive surfaces to generate a prescribed intensity distribution,” Comput. Opt. 39(4), 508–514 (2015).
[Crossref]

M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23, A1140–A1148 (2015).
[Crossref]

Mashaal, H.

Moiseev, M. A.

S. V. Kravchenko, M. A. Moiseev, E. V. Byzov, and L. L. Doskolovich, “Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions,” Opt. Commun. 459, 124976 (2020).
[Crossref]

S. V. Kravchenko, E. V. Byzov, M. A. Moiseev, and L. L. Doskolovich, “Design of optical elements with two refractive surfaces to generate a prescribed intensity distribution,” Comput. Opt. 39(4), 508–514 (2015).
[Crossref]

M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23, A1140–A1148 (2015).
[Crossref]

M. A. Moiseev and L. L. Doskolovich, “Design of TIR optics generating the prescribed irradiance distribution in the circle region,” J. Opt. Soc. Am. A 29, 1758–1763 (2012).
[Crossref]

Qureshi, M. A.

Robbie, S. J.

Silva-Lora, A.

Singer, W.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

Tabernero, J.

Torres, R.

Totzeck, M.

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

Tyrkevich, Y. G.

S. T. Bobrov, G. I. Greisukh, and Y. G. Tyrkevich, Optics of Diffractive Elements and Systems (Mashinostroenie, 1986).

Wolf, E.

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1959).

Zeng, L.

Appl. Opt. (4)

Biomed. Opt. Express (2)

Comput. Opt. (1)

S. V. Kravchenko, E. V. Byzov, M. A. Moiseev, and L. L. Doskolovich, “Design of optical elements with two refractive surfaces to generate a prescribed intensity distribution,” Comput. Opt. 39(4), 508–514 (2015).
[Crossref]

J. Opt. Soc. Am. A (4)

Opt. Commun. (2)

R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutierrez-Vega, “Exact equations for stigmatic singlet design meeting the Abbe sine condition,” Opt. Commun. 479, 126415 (2021).
[Crossref]

S. V. Kravchenko, M. A. Moiseev, E. V. Byzov, and L. L. Doskolovich, “Design of axisymmetric double-surface refractive optical elements generating required illuminance distributions,” Opt. Commun. 459, 124976 (2020).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Other (4)

H. Gross, W. Singer, M. Totzeck, F. Blechinger, and B. Achtner, Handbook of Optical Systems (Wiley, 2005).

M. Born and E. Wolf, Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1959).

https://www.zemax.com/ .

S. T. Bobrov, G. I. Greisukh, and Y. G. Tyrkevich, Optics of Diffractive Elements and Systems (Mashinostroenie, 1986).

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (4)

Fig. 1.
Fig. 1. Geometry of the lens design problem.
Fig. 2.
Fig. 2. (a) Function $\gamma (\phi)$ calculated from the solution of the differential Eq. (12) (solid red line) and its linear approximation (dashed blue line). (b) Designed lens imported into the Zemax software. (c) Cross-section of the point spread function in the image plane calculated using Zemax.
Fig. 3.
Fig. 3. Fresnel losses on two lens surfaces versus angle $\phi$ for the designed lens (solid red line) and for the plano-convex stigmatic lens (solid blue line). The dashed lines show the Fresnel losses for the same lenses with $\lambda /4$ antireflective coatings.
Fig. 4.
Fig. 4. Function ${\delta _{{\rm{mag}}}}(\phi)$ describing the normalized variation of the magnification of the designed lens.

Equations (30)

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{ z 2 ( ϕ ) = R ( ϕ ) cos ϕ + l ( ϕ ) cos γ ( ϕ ) , y 2 ( ϕ ) = R ( ϕ ) sin ϕ + l ( ϕ ) sin γ ( ϕ ) .
Ψ 0 = R ( ϕ ) + n l ( ϕ ) + y 2 2 ( ϕ ) + ( f z 2 ( ϕ ) ) 2 = c o n s t .
Ψ 0 = R 0 + n l 0 + ( f [ R 0 + l 0 ] ) = ( n 1 ) l 0 + f ,
R ( ϕ ) = Ψ 0 2 f 2 + l ( ϕ ) [ 2 n Ψ 0 + 2 f cos γ ( ϕ ) + ( n 2 1 ) l ( ϕ ) ] 2 [ Ψ 0 f cos ϕ + ( cos ( ϕ γ ( ϕ ) ) n ) l ( ϕ ) ] .
β ( ϕ ) = 2 γ ( ϕ ) ϕ .
sin ( β ( ϕ ) ) = sin ( ϕ 2 γ ( ϕ ) ) = y 2 ( ϕ ) y 2 2 ( ϕ ) + ( f z 2 ( ϕ ) ) 2 = y 2 ( ϕ ) Ψ 0 R ( ϕ ) n l ( ϕ ) .
l ( ϕ ) = Ψ 0 cos ( ϕ γ ( ϕ ) ) f cos γ ( ϕ ) 1 + n cos ( ϕ γ ( ϕ ) ) .
R ( ϕ ) = Ψ 0 + f [ n sin ( ϕ 2 γ ( ϕ ) ) + sin γ ( ϕ ) ] csc ( ϕ γ ( ϕ ) ) 2 + 2 n cos ( ϕ γ ( ϕ ) ) .
Ψ ( z 1 ( ϕ ) , y 1 ( ϕ ) ) = Ψ ( R ( ϕ ) cos ϕ , R ( ϕ ) sin ϕ ) = R ( ϕ ) .
n cos ( γ ( ϕ ) ) d d ϕ [ R ( ϕ ) cos ϕ ] + n sin ( γ ( ϕ ) ) d d ϕ [ R ( ϕ ) sin ϕ ] = d R ( ϕ ) d ϕ .
d ln R ( ϕ ) d ϕ = sin ( ϕ γ ( ϕ ) ) 1 / n cos ( ϕ γ ( ϕ ) ) .
d γ ( ϕ ) d ϕ = F ( ϕ , γ ( ϕ ) ) = 4 f [ n cos ( ϕ γ ( ϕ ) ) ] [ 1 + n cos ( ϕ γ ( ϕ ) ) ] sin γ ( ϕ ) 4 f ( 1 + n 2 ) sin ϕ n [ 4 Ψ 0 sin 3 ( ϕ γ ( ϕ ) ) + f ξ ( γ , ϕ ) ] ,
F ( 0 , γ ( 0 ) ) = lim ϕ 0 F ( ϕ , γ ( ϕ ) ) = α .
R 0 = f ( n + α 2 n α ) + ( α 1 ) Ψ 0 2 ( n 1 ) ( α 1 ) .
α = 1 + f ( n 1 ) f 2 f n + 2 ( n 1 ) R 0 + Ψ 0 = 1 f 2 f 2 R 0 l 0 .
R ( ϕ ) = R 0 [ n 1 n cos ( ϕ α ϕ ) 1 ] 1 1 α , l ( ϕ ) = Ψ 0 cos ( ϕ α ϕ ) f cos ( α ϕ ) 1 + n cos ( ϕ α ϕ ) .
R ( ϕ ) = R 0 n 1 n cos ϕ 1 , l ( ϕ ) = Ψ 0 cos ϕ f n cos ϕ 1 .
r 1 = R 0 f 2 f n + 2 ( n 1 ) R 0 + Ψ 0 f ( n 1 ) + 2 ( n 1 ) R 0 + Ψ 0 , r 2 = ( f n ( n 1 ) R 0 Ψ 0 ) ( f + 2 ( n 1 ) R 0 + Ψ 0 ) ( n 1 ) ( f + f n 2 ( n 1 ) R 0 Ψ 0 ) .
L m a g = sin ϕ sin β ( ϕ ) = 1 2 α 1 + O ( ϕ 2 ) 1 2 f 2 R 0 + l 0 , A m a g = tan β ( ϕ ) tan ϕ sin β ( ϕ ) sin ϕ = 1 L m a g .
γ ( ϕ ) = α ϕ + σ ϕ 3 + O ( ϕ 5 ) ,
F ( 0 , γ ( 0 ) ) + 3 σ ϕ 2 + O ( ϕ 4 ) F ( ϕ , α ϕ + σ ϕ 3 + O ( ϕ 5 ) ) = 0.
[ 2 σ + α ( 1 α ) ( 1 2 α ) 3 l 0 n α ( 1 α ) 3 f ( n 1 ) ] ϕ 2 + O ( ϕ 4 ) .
σ = 1 6 α ( 1 α ) ( 2 α 1 ) + n 2 ( n 1 ) l 0 f α ( 1 α ) 3 .
L m a g = γ 0 + γ 2 ϕ 2 + O ( ϕ 4 ) ,
γ 0 = 1 2 α 1 , γ 2 = 1 6 ( 2 α 1 ) + ( 2 α 1 ) 3 12 σ 6 ( 2 α 1 ) 2 .
γ 0 = γ 0 ( R 0 , l 0 , f ) = L m a g , γ 2 = γ 2 ( R 0 , l 0 , f ) = 0.
R 0 = f n + L m a g ( n 2 ) n ( L m a g 1 ) 2 , l 0 = 4 f L m a g n 1 n ( L m a g 1 ) 2 .
E = [ 1 1 E 0 0 ϕ m a x F ( ϕ ) sin ϕ d ϕ ] × 100 % ,
sin ϕ sin β ( ϕ ) = c o n s t = L m a g .
δ m a g ( ϕ ) = 1 | L m a g | [ sin ϕ sin ( 2 γ ( ϕ ) ϕ ) L m a g ]