Abstract

We derive the analytic formula of the output surface of a spherochromatic lens. The analytic solution ensures that all the rays for a wide range of wavelengths fall inside the Airy disk. So, its amount of spherical aberration is small enough to consider the lens as diffracted limited. We test the singlet lens using ray-tracing methods and find satisfactory results, including spot diagram analysis for three different Abbe wavelengths.

© 2019 Optical Society of America

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References

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  1. M. Bass, Handbook of Optics, Volume I: Fundamentals, Techniques, and Design (McGraw-Hill, Inc., 1995).
  2. H. Sun, Lens Design: A Practical Guide (CRC Press, 2016).
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  8. F. J. Dijksterhuis, Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer Science & Business Media, 2004), Vol. 9.
  9. W. P. McCray, Stargazer: the Life and Times of the Telescope (2006).
  10. 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]
  11. G. Schulz, “Achromatic and sharp real imaging of a point by a single aspheric lens,” Appl. Opt. 22, 3242–3248 (1983).
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    [Crossref]
  13. J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).
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    [Crossref]
  15. R. G. González-Acuña and J. C. Guitiérrez-Vega, “Generalization of the axicon shape: the gaxicon,” J. Opt. Soc. Am. A 35, 1915–1918 (2018).
    [Crossref]
  16. R. G. González-Acuña, H. A. Chaparro-Romo, and J. C. Gutiérrez-Vega, “General formula to design freeform singlet free of spherical aberration and astigmatism,” Appl. Opt. 58, 1010–1015 (2019).
    [Crossref]
  17. R. G. González-Acuña, M. Avendaño-Alejo, and J. C. Gutiérrez-Vega, “Singlet lens for generating aberration-free patterns on deformed surfaces,” J. Opt. Soc. Am. A 36, 925–929 (2019).
    [Crossref]
  18. O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
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2019 (2)

2018 (3)

2012 (1)

2004 (1)

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

1983 (1)

Avendaño-Alejo, M.

Bass, M.

M. Bass, Handbook of Optics, Volume I: Fundamentals, Techniques, and Design (McGraw-Hill, Inc., 1995).

Benitez, P.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Bentez, P.

Blen, J.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Braunecker, B.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press, 2008), Vol. 173.

Campbell, S.

Chaparro-Romo, H. A.

Chaves, J.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).

Daumas, M.

M. Daumas, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, London (UK) (Batsford, 1972).

Dijksterhuis, F. J.

F. J. Dijksterhuis, Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer Science & Business Media, 2004), Vol. 9.

Dross, O.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Duerr, F.

González-Acuña, R. G.

Guitiérrez-Vega, J. C.

Gutiérrez-Vega, J. C.

Hentschel, R.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press, 2008), Vol. 173.

Hernandez, M.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Huygens, C.

C. Huygens, Traité de la lumière (Macmillan, 1920).

McCray, W. P.

W. P. McCray, Stargazer: the Life and Times of the Telescope (2006).

Meuret, Y.

Minano, J. C.

F. Duerr, P. Bentez, J. C. Minano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express 20, 5576–5585 (2012).
[Crossref]

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Mohedano, R.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Munoz, F.

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Nagar, J.

Newton, I.

I. Newton, Opticks, or, a Treatise of the Reflections, Refractions, Inflections & Colours of Light (Courier Corporation, 1979).

Schulz, G.

Sun, H.

H. Sun, Lens Design: A Practical Guide (CRC Press, 2016).

Thienpont, H.

Tiziani, H. J.

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press, 2008), Vol. 173.

Toomer, G. J.

G. J. Toomer, Diocles, On Burning Mirrors: The Arabic Translation of the Lost Greek Original (Springer Science & Business Media, 2012), Vol. 1.

Werner, D.

Appl. Opt. (3)

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

Opt. Express (1)

Optica (1)

Proc. SPIE (1)

O. Dross, R. Mohedano, P. Benitez, J. C. Minano, J. Chaves, J. Blen, M. Hernandez, and F. Munoz, “Review of SMS design methods and real-world applications,” Proc. SPIE 5529, 35–48 (2004).
[Crossref]

Other (10)

J. Chaves, Introduction to Nonimaging Optics, 2nd ed. (CRC Press, 2016).

M. Bass, Handbook of Optics, Volume I: Fundamentals, Techniques, and Design (McGraw-Hill, Inc., 1995).

H. Sun, Lens Design: A Practical Guide (CRC Press, 2016).

B. Braunecker, R. Hentschel, and H. J. Tiziani, Advanced Optics Using Aspherical Elements (SPIE Press, 2008), Vol. 173.

I. Newton, Opticks, or, a Treatise of the Reflections, Refractions, Inflections & Colours of Light (Courier Corporation, 1979).

M. Daumas, Scientific Instruments of the Seventeenth and Eighteenth Centuries and Their Makers, London (UK) (Batsford, 1972).

G. J. Toomer, Diocles, On Burning Mirrors: The Arabic Translation of the Lost Greek Original (Springer Science & Business Media, 2012), Vol. 1.

C. Huygens, Traité de la lumière (Macmillan, 1920).

F. J. Dijksterhuis, Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (Springer Science & Business Media, 2004), Vol. 9.

W. P. McCray, Stargazer: the Life and Times of the Telescope (2006).

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Figures (5)

Fig. 1.
Fig. 1. Meridional half-plane of the spherochromatic singlet, illustrating the deviation of the incoming rays by the surfaces and their optical paths traversed. For clarity, we show only two rays, but the analysis is general for any number of wavelengths, and thus we use the notation $ {h_i} $ for the $ i $-th ray.
Fig. 2.
Fig. 2. (a) Surface of the singlet $ {z_b}({r_b}) $ for the parameters given in the text. Solid black line is the numeric solution of the exact system of Eqs. (1) and (4). Dashed red line is the plot of the analytic solution (5). (b) Percentage error between both solutions.
Fig. 3.
Fig. 3. Ray diagram of a spherochromatic singlet lens for light with $ {\lambda _1} = 587.6\,\,{\rm nm} $. The ray trajectories for $ {\lambda _2} $ and $ {\lambda _3} $ have been omitted because they practically overlap with rays with $ {\lambda _1} $, making it difficult to visualize. Geometrical parameters are included within the text.
Fig. 4.
Fig. 4. Percentage similitude of the optical paths (PSOP). Black PSOP curve is for a stigmatic lens. Gray line is the sum of each optical path of Eq. (1). Red, orange, and yellow curves represent the PSOP curves for wavelengths $ {\lambda _1} $, $ {\lambda _2} $, $ {\lambda _3} $, respectively.
Fig. 5.
Fig. 5. Single-spot diagrams for the three wavelengths $ {\lambda _1} $, $ {\lambda _2} $, and $ {\lambda _3} $. Black circles correspond to the Airy disks. For all the wavelengths, the spot images are inside the Airy disks.

Equations (17)

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t a + n i t + t b = ( r a + h i ) 2 + t a 2 + n i z b 2 + ( r b r a h i ) 2 + r b 2 + ( z b t t b ) 2 ,
z ^ = [ 1 , 0 ]
u ^ i = [ t a , r a + h i ] ( r a + h i ) 2 + t a 2 ,
v ^ i = [ z b , r b r a h i ] ( r b r a h i ) 2 + z b 2
v ^ i = z ^ n i × ( z ^ × u ^ i ) + z ^ n i n i 2 | u ^ i × z ^ | 2 .
R i r b r a h i ( r b r a h i ) 2 + z b 2 = r a + h i n i ( r a + h i ) 2 + t a 2 ,
Z i z b ( r b r a h i ) 2 + z b 2 = 1 ( r a + h i ) 2 n i 2 [ ( r a + h i ) 2 + t a 2 ] ,
z b = Z 1 A B + C + D A ,
r b = r a + R 1 z b Z 1 ,
h i = n i 2 R 1 2 t a 2 n i R 1 t a Z 1 2 ( n i 2 1 ) R 1 2 r a ,
A Z 1 2 [ 9 ( i = 1 3 n i Z i ) 2 ] + 9 R 1 2 ,
B ( F 3 t b 3 t ) ( F + 3 t b + 3 t ) 9 r a 2 ,
C { Z 1 [ F i = 1 3 n i Z i 9 ( t b + t ) ] + 9 R 1 r a } 2 ,
D 9 Z 1 [ Z 1 ( t b + t ) R 1 r a ] F Z 1 2 i = 1 3 n i Z i ,
F t i = 1 3 n i + 3 t b ( 3 t a i = 1 3 ( r a + h i ) 2 + t a 2 ) .
ε = | E x a c t A p p r o x | | E x a c t | × 100 %
( 1 | L S H o f E q . ( 1 ) R S H o f E q . ( 1 ) L S H o f E q . ( 1 ) | ) × 100 % ,

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