Abstract

In this paper, we present a rigorous analytical solution for the bi-aspheric singlet lens design problem. The input of the general formula presented here is the first surface of the singlet lens; this surface must be continuous and such that the rays inside the lens do not cross each other. The output is the correcting second surface of the singlet; the second surface is such that the singlet is free of spherical aberration.

© 2018 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
General formula to design a freeform singlet free of spherical aberration and astigmatism

Rafael G. González-Acuña, Héctor A. Chaparro-Romo, and Julio C. Gutiérrez-Vega
Appl. Opt. 58(4) 1010-1015 (2019)

Singlet lens for generating aberration-free patterns on deformed surfaces

Rafael G. González-Acuña, Maximino Avendaño-Alejo, and Julio C. Gutiérrez-Vega
J. Opt. Soc. Am. A 36(5) 925-929 (2019)

References

  • View by:
  • |
  • |
  • |

  1. D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design (CRC Press, 2016).
  2. R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).
  3. D. Malacara, “Two lenses to collimate red laser light,” Appl. Opt. 4, 1652–1654 (1965).
    [Crossref]
  4. E. M. Vaskas, “Note on the Wasserman–Wolf method for designing aspheric surfaces,” J. Opt. Soc. Am. 47, 669–670 (1957).
    [Crossref]
  5. G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. London Sect. B 62, 2–8 (1949).
    [Crossref]
  6. P. D. Lin and C.-Y. Tsai, “Determination of unit normal vectors of aspherical surfaces given unit directional vectors of incoming and outgoing rays: reply,” J. Opt. Soc. Am. A 29, 1358 (2012).
    [Crossref]
  7. J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
    [Crossref]
  8. J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
    [Crossref]
  9. M. Avendaño-Alejo, E. Román-Hernández, L. Castañeda, and V. I. Moreno-Oliva, “Analytic conic constants to reduce the spherical aberration of a single lens used in collimated light,” Appl. Opt. 56, 6244–6254 (2017).
    [Crossref]
  10. G. Castillo-Santiago, M. Avendaño-Alejo, R. Daz-Uribe, and L. Castañeda, “Analytic aspheric coefficients to reduce the spherical aberration of lens elements used in collimated light,” Appl. Opt. 53, 4939–4946 (2014).
    [Crossref]
  11. N. D. C. Lozano-Rincón and J. C. Valencia-Estrada, “Paraboloid-aspheric lenses free of spherical aberration,” J. Modern Opt. 64, 1146–1157 (2017).
    [Crossref]
  12. J. C. Valencia-Estrada, J. Garca-Marquez, L. Chassagne, and S. Topsu, “Catadioptric interfaces for designing VLC antennae,” Appl. Opt. 56, 7559–7566 (2017).
    [Crossref]
  13. R. G. González-Acuña and H. A. Chaparro-Romo, “Singlet free of spherical aberration,” 2018, https://doi.org/10.6084/m9.figshare.7163357 .

2017 (4)

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

N. D. C. Lozano-Rincón and J. C. Valencia-Estrada, “Paraboloid-aspheric lenses free of spherical aberration,” J. Modern Opt. 64, 1146–1157 (2017).
[Crossref]

M. Avendaño-Alejo, E. Román-Hernández, L. Castañeda, and V. I. Moreno-Oliva, “Analytic conic constants to reduce the spherical aberration of a single lens used in collimated light,” Appl. Opt. 56, 6244–6254 (2017).
[Crossref]

J. C. Valencia-Estrada, J. Garca-Marquez, L. Chassagne, and S. Topsu, “Catadioptric interfaces for designing VLC antennae,” Appl. Opt. 56, 7559–7566 (2017).
[Crossref]

2015 (1)

J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
[Crossref]

2014 (1)

2012 (1)

1965 (1)

1957 (1)

1949 (1)

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. London Sect. B 62, 2–8 (1949).
[Crossref]

Avendaño-Alejo, M.

Castañeda, L.

Castillo-Santiago, G.

Chaparro-Romo, H. A.

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

Chassagne, L.

Daz-Uribe, R.

Flores-Hernández, R. B.

J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
[Crossref]

Garca-Marquez, J.

Herzberger, M.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Lin, P. D.

Lozano-Rincón, N. D. C.

N. D. C. Lozano-Rincón and J. C. Valencia-Estrada, “Paraboloid-aspheric lenses free of spherical aberration,” J. Modern Opt. 64, 1146–1157 (2017).
[Crossref]

Luneburg, R. K.

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

Malacara, D.

Malacara-Hernández, D.

J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
[Crossref]

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design (CRC Press, 2016).

Malacara-Hernández, Z.

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design (CRC Press, 2016).

Moreno-Oliva, V. I.

Pereira-Ghirghi, M. V.

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

Román-Hernández, E.

Topsu, S.

Tsai, C.-Y.

Valencia-Estrada, J. C.

N. D. C. Lozano-Rincón and J. C. Valencia-Estrada, “Paraboloid-aspheric lenses free of spherical aberration,” J. Modern Opt. 64, 1146–1157 (2017).
[Crossref]

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

J. C. Valencia-Estrada, J. Garca-Marquez, L. Chassagne, and S. Topsu, “Catadioptric interfaces for designing VLC antennae,” Appl. Opt. 56, 7559–7566 (2017).
[Crossref]

J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
[Crossref]

Vaskas, E. M.

Wassermann, G.

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. London Sect. B 62, 2–8 (1949).
[Crossref]

Wolf, E.

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. London Sect. B 62, 2–8 (1949).
[Crossref]

Appl. Opt. (4)

J. Modern Opt. (1)

N. D. C. Lozano-Rincón and J. C. Valencia-Estrada, “Paraboloid-aspheric lenses free of spherical aberration,” J. Modern Opt. 64, 1146–1157 (2017).
[Crossref]

J. Opt. (1)

J. C. Valencia-Estrada, M. V. Pereira-Ghirghi, Z. Malacara-Hernández, and H. A. Chaparro-Romo, “Aspheric coefficients of deformation for a Cartesian oval surface,” J. Opt. 46, 100–107 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

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

Proc. Phys. Soc. London Sect. B (1)

G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. London Sect. B 62, 2–8 (1949).
[Crossref]

Proc. R. Soc. A (1)

J. C. Valencia-Estrada, R. B. Flores-Hernández, and D. Malacara-Hernández, “Singlet lenses free of all orders of spherical aberration,” Proc. R. Soc. A 471, 20140608 (2015).
[Crossref]

Other (3)

D. Malacara-Hernández and Z. Malacara-Hernández, Handbook of Optical Design (CRC Press, 2016).

R. K. Luneburg and M. Herzberger, Mathematical Theory of Optics (University of California, 1964).

R. G. González-Acuña and H. A. Chaparro-Romo, “Singlet free of spherical aberration,” 2018, https://doi.org/10.6084/m9.figshare.7163357 .

Supplementary Material (1)

NameDescription
» Code 1       mathematica code

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Diagram of a singlet lens free of spherical aberration. The first surface is given by ( r a , z a ) , and the second surface is given by ( r b , z b ) . The distance between the first surface and the object is t a , the thickness at the center of the lens is t , and the distance between the second surface and the image is t b .
Fig. 2.
Fig. 2. Left: Zoom at point P of Fig. (1); three unitary vectors can be seen: v 1 is the unitary vector of the incident ray, v 2 is the unitary vector of the refracted ray, and n a is the normal vector of the first surface. Right: Zoom at point Q of Fig. (1); unitary vectors of the second surface can be seen: v 2 is the unitary vector of the incited ray, v 3 is the unitary vector of the refracted ray, and n b is the normal vector of the second surface.
Fig. 3.
Fig. 3. Gallery of singlet lenses bi-aspherical free of spherical aberration. For all cases, the configuration is t a = 70    mm , t = 8    mm , t b = 75    mm , and n = 1.5 .
Fig. 4.
Fig. 4. (a)  t b = 75    mm , (b)  t b = 55    mm , (c)  t b = 35    mm , (d)  t b = 15    mm . The constant configuration of cases (a)–(d) is t a 30    mm , t = 5    mm , n = 1.5 .
Fig. 5.
Fig. 5. Configuration of the singlet lens with negative refraction index: t a = 20    mm , t = 3    mm , t b = 30    mm , n = 1.5 .
Fig. 6.
Fig. 6. Configuration of the singlet lens with object at infinity: t = 3.5    mm , t b = 50    mm , n = 1.5 .

Tables (2)

Tables Icon

Table 1. Benchmark of Resolver Power Between Analytical Solutions to Design of Singlet Lenses Free of Spherical Aberration

Tables Icon

Table 2 Singlet free of spherical aberration.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

v 2 = 1 n [ n a × ( n a × v 1 ) ] n a 1 1 n 2 ( n a × v 1 ) · ( n a × v 1 ) ,
{ v 1 = [ r a , ( z a t a ) , 0 ] r a 2 + ( z a t a ) 2 , v 2 = [ r b r a , z b z a , 0 ] ( r b r a ) 2 + ( z b z a ) 2 , n a = [ z a , 1 , 0 ] 1 + z a 2 ,
{ r i r b r a Ψ = ( z a t a ) z a + r a n r a 2 + ( t a z a ) 2 ( 1 + z a 2 ) z a ϕ , z i z b z a Ψ = ( r a + ( z a t a ) z a ) z a n r a 2 + ( t a z a ) 2 ( 1 + z a 2 ) + ϕ , with , Ψ = ( z b z a ) 2 + ( r b r a ) 2 ϕ = 1 ( r a + ( z a t a ) z a ) 2 n 2 ( r a 2 + ( t a z a ) 2 ) ( 1 + z a 2 ) 1 + z a 2
t a + n t + t b = sgn ( t a ) r a 2 + ( z a t a ) 2 + n ( r b r a ) 2 + ( z b z a ) 2 + sgn ( t b ) r b 2 + ( z b t t b ) 2 ,
{ r b = r i ( z b z a ) z i + r a , z b = h 0 + s 1 z i 2 [ 2 n f i ( z i ( z a t b + t ( z i 1 ) ) + r a r i + t r i 2 ) ( r i ( z a + t b + t ) + r a z i ) 2 + f i 2 + h 1 n 2 ] 1 n 2 ,
{ f i = sgn ( t a ) r a 2 + ( t a z a ) 2 + t a t b , h 0 = n f i z i n 2 ( t z i + z a ) + r i 2 z a r a r i z i + z i 2 ( t + t b ) , h 1 = r a 2 + 2 r a r i t + ( t b z a ) 2 + t 2 ( r i 2 + ( 1 + z i ) 2 ) 2 t ( t b z a ) ( 1 + z i ) .
Ω a = { ( r a , z a ) R 2 | z a < z b } , Ω b = { ( r b , z b ) R 2 | z b > z a } ,
RS Error = N ( z b z b c ) 2 N ,
{ a )    z a = { exp ( r a ) + J 0 ( r a ) with , r a < 0 exp ( r a ) + J 0 ( r a ) with , r a 0 , b )    z a = 1 18 r a 2 + cos ( r a 2 ) , c )    z a = 5 cos ( r a 5 ) , d )    z a = 1 18 r a 2 + cos ( r a 2 ) ,
{ lim t a f i = z a t b , lim t a z i = ( n 2 1 ) z a 2 + n 2 n 2 ( z a 2 + 1 ) z a 2 + 1 + z a 2 n z a 2 + n , lim t a r i = z a ( n z a 2 + 1 ( n 2 1 ) z a 2 + n 2 n 2 ( z a 2 + 1 ) 1 ) n ( z a 2 + 1 ) .
{ n b = [ z b , r b , 0 ] z b 2 + r b 2 , v 3 = [ r b , z b t t b , 0 ] r b 2 + ( z b t t b ) 2 , v 3 = n [ n b × ( n b × v 2 ) ] n b 1 n 2 ( n b × v 2 ) · ( n b × v 2 ) .
E = 100 % | v 3 v 3 v 3 | × 100 % .

Metrics