Abstract

Lens design uses a calculation of the lens’ surfaces that permits us to obtain an image from a given object. A set of general rules and laws permits us to calculate the essential points of the optical system, such as distances, thickness, pupils, and focal distances, among others. Now, the theory on which classical lens design is based has changed radically, as our theoretical foundations do not rely on the classical ray-tracing rules. We show that with the rules expressed in a reduced vector analytical solution set of equations, we can take into account all optical elements, i.e., refractive, reflective, and catadioptric. These foundations permit us to keep under control the system aberration budget in every surface. It reduces the computation time dramatically. The examples presented here were possible because of the versatility of this theoretical approach.

© 2019 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2012).
  2. O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-Function and its Ramifications (Wiley-VCH, 2006).
  3. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).
  4. A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, 1985). (First published by Oxford University Press, London, in 1929).
  5. D. Malacara and Z. Malacara, Handbook of Optical Design, 3rd ed. (CRC Press, 2013).
  6. W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).
  7. H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005), Vol. 1: Fundamentals of Technical Optics.
  8. R. K. Luneburg, Mathematical Theory of Optics (University of California, 1964).
  9. A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
    [Crossref]
  10. Y. Nie, H. Gross, Y. Zhong, and F. Duerr, “Freeform optical design for a nonscanning corneal imaging system with a convexly curved image,” Appl. Opt. 56, 5630–5638 (2017).
    [Crossref]
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    [Crossref]
  12. J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17, 24036–24044 (2009).
    [Crossref]
  13. P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
    [Crossref]

2018 (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

2017 (1)

2010 (1)

2009 (1)

2004 (1)

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Arroyo, R. M.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Bauer, A.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Benítez, P.

Blen, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2012).

Chaves, J.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Cheng, D.

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, 1985). (First published by Oxford University Press, London, in 1929).

Dross, O.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Duerr, F.

Falicoff, W.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Gimenez-Benitez, P.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Gross, H.

Hernandez, M.

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Hua, H.

Infante, J.

Lin, W.

Luneburg, R. K.

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

Malacara, D.

D. Malacara and Z. Malacara, Handbook of Optical Design, 3rd ed. (CRC Press, 2013).

Malacara, Z.

D. Malacara and Z. Malacara, Handbook of Optical Design, 3rd ed. (CRC Press, 2013).

Miñano, J. C.

J. C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express 17, 24036–24044 (2009).
[Crossref]

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Muñoz, F.

Nie, Y.

Rolland, J. P.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Santamaría, A.

Sasian, J.

Schiesser, E. M.

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

Stavroudis, O. N.

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-Function and its Ramifications (Wiley-VCH, 2006).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

Wang, Y.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2012).

Zhong, Y.

Appl. Opt. (1)

Nat. Commun. (1)

A. Bauer, E. M. Schiesser, and J. P. Rolland, “Starting geometry creation and design method for free-form optics,” Nat. Commun. 9, 1756 (2018).
[Crossref]

Opt. Eng. (1)

P. Gimenez-Benitez, J. C. Miñano, J. Blen, R. M. Arroyo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43, 1489–1503 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Other (8)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge University, 2012).

O. N. Stavroudis, The Mathematics of Geometrical and Physical Optics: The k-Function and its Ramifications (Wiley-VCH, 2006).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, 1972).

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover Publications, 1985). (First published by Oxford University Press, London, in 1929).

D. Malacara and Z. Malacara, Handbook of Optical Design, 3rd ed. (CRC Press, 2013).

W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2007).

H. Gross, Handbook of Optical Systems (Wiley-VCH, 2005), Vol. 1: Fundamentals of Technical Optics.

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

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Figures (6)

Fig. 1.
Fig. 1. Current and proposed optical variables in ray tracing. In (a) we represent the current way to describe all the variables that permits trace rays for image formation. Note that the chief and marginal rays describe the position of the inverted image and that the optical axis lies on a straight line that crosses the lens by its vertex. In (b) is the novel optical path tracing describing both the reference and arbitrary optical paths related to an arbitrary origin of coordinates. Note that the image should not be necessarily inverted, as it could meet the specific optical system’s requirements. The optical axis does not lie on a straight line but is deliberately tilted to show the method’s performance. All paths are vectored.
Fig. 2.
Fig. 2. Lens presenting an internal fan of rays crossing internally. The vector formulas for freeform optical system design are useful to design on axis systems with a collinear reference optical path.
Fig. 3.
Fig. 3. Two different objects’ positions are depicted here: ${P_0}$ and $ {P_0^\prime } $ , with their respective images ${P_3}$ and $ {P_3^\prime } $ . We can see the external reference optical paths following points $ {P_0^\prime } $ to $ {P_3^\prime } $ . The external reference paths allow us to indirectly measure aberrations in the image by calculating the ideal normal $ {{\textbf{n}_2^\prime }} $ in $ {P_2^\prime } $ . On the opposite side, only one main reference optical path ( $\overline{P_0O_1O_2P_3}$ ) is needed to design the lens. The lens has two convex surfaces ${S_1}$ and ${S_2}$ . The wavefront $W$ entering the first surface is also depicted. Note that both surfaces are not parallel, but tilted, and their thickness is larger at the bottom, in turn introducing a prism. The refractive index between surfaces ${S_1}$ and ${S_2}$ is ${n_1}$ , and ${n_2} = {n_0}$ . The versatility of this designing method permits the introduction of tilt or change of the distribution of the image.
Fig. 4.
Fig. 4. Example of a freeform lens designed with an external reference path. The off-axis lens has an off-axis reference aperture’s domain, and the lens’s surfaces ${S_1}$ and ${S_2}$ are highly tilted between them. In this example, ${S_1}$ is an off-axis section of a revolution paraboloid. The aperture’s domain rim defines the surface’s edges. The reference optical path, which consists of three segments, is visualized with phantom lines.
Fig. 5.
Fig. 5. More complex example of catadioptric design with the proposed method in which object ${\boldsymbol{O}}$ is virtual. The aperture limits the entrance of converging rays into the optical system. The second surface is not refractive, but reflective, and it deviates the rays into image ${\boldsymbol{I}}$ . The extended reference optical path is represented with phantom lines.
Fig. 6.
Fig. 6. Freeform optical system. (a) A single prismatic and refractive two-surface optical system images an LED. Its ${S_2}$ surface was calculated by using Eq. (18). (b) This surface ${S_2}$ is shown in green (the parametric one), and, in red, its best fitting (explicit one) is shown. The explicit surface was fitted with Zernike’s polynomials. (c) After merging both surfaces, its difference is plotted in a cloud of uniformly distributed sample points. Note the insignificance of the residual error (expressed in millimeters).

Equations (41)

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p 0 = [ x 0 , y 0 , z 0 ] p 1 = [ x 1 , y 1 , z 1 ] p 2 = [ x 2 , y 2 , z 2 ] p 3 = [ x 3 , y 3 , z 3 ] } .
a 0 = p 1 p 0 a 1 = p 2 p 1 a 2 = p 3 p 2 } ,
r 0 = o 1 p 0 r 1 = o 2 o 1 r 2 = p 3 o 2 } .
z 1 | O 1 = τ ( ± n 1 r 1 / r 1 n 0 r 0 / r 0 ) .
± n 0 a 0 + n 1 a 1 ± n 2 a 2 = ± n 0 r 0 + n 1 r 1 ± n 2 r 2 = K .
Sign ( r 0 r 1 ) n 0 a 0 + n 1 a 1 + Sign ( r 1 r 2 ) n 2 a 2 = Sign ( r 0 r 1 ) n 0 r 0 + n 1 r 1 + Sign ( r 1 r 2 ) n 2 r 2 = K ,
v 0 = Sign ( r 0 r 1 ) a 0 a 0 a 0 .
n 1 = p 1 x × p 1 y p 1 x × p 1 y | x y .
v 1 = n 0 n 1 ( v 0 ( n 1 v 0 ) n 1 ) ( 1 n 0 2 n 1 2 ( 1 ( n 1 v 0 ) 2 ) ) n 1 = [ a b c ] ,
Sign ( r 1 r 2 ) n 2 a 2 v 1 = Sign ( r 0 r 1 ) n 0 r 0 v 1 + n 1 r 1 v 1 + Sign ( r 1 r 2 ) n 2 r 2 v 1 Sign ( r 0 r 1 ) n 0 a 0 v 1 n 1 a 1 v 1 .
( ( p 3 p 2 ) ( p 3 p 2 ) ) v 1 = ( k 1 v 1 n 1 n 2 ( p 2 p 1 ) ) 2 ,
( p 2 p 2 ) v 1 2 2 ( p 2 p 3 ) v 1 2 = ( k 1 v 1 n 1 n 2 ( p 2 p 1 ) ) 2 ( p 3 p 3 ) v 1 2 .
( p 2 p 2 ) v 1 2 = p 2 2 2 p 2 ( p 1 ( p 1 v 1 ) v 1 ) + p 1 2 + ( p 1 p 1 ) v 1 2 2 ( p 1 v 1 ) p 1 v 1 .
2 ( p 2 p 3 ) v 1 2 = 2 ( ( p 3 v 1 ) ( p 2 p 1 ) v 1 + ( p 1 p 3 ) v 1 2 ) .
( n 2 2 n 1 2 n 2 ) a 1 2 2 k 3 v 1 a 1 + n 2 ( k 2 k 1 2 ) v 1 2 = 0 ,
a 1 = n 2 n 2 2 n 1 2 ( k 3 + Sign ( r 1 r 2 ) k 3 2 ( n 2 2 n 1 2 ) ( k 2 k 1 2 ) ) v 1 .
p 2 = p 1 + a 1 = p 1 + n 2 n 2 2 n 1 2 × ( k 3 + Sign ( r 1 r 2 ) ( k 3 2 ( n 2 2 n 1 2 ) ( k 2 k 1 2 ) ) ) v 1 .
p 2 x × p 2 y | x y [ 0 , 0 , 0 ]     ( x , y ) Aperture,
p 2 = p 1 + a 1 = p 1 + ( n 2 G V s 3 V 2 ( n 2 2 n 1 2 ) G ) v 1 .
p 2 = p 1 + a 1 = p 1 + ( n 2 G 2 V ) v 1 .
A = k 1 = s 0 ( n 0 / n 2 ) ( r 0 a 0 ) + ( n 1 / n 2 ) r 1 + s 2 r 2 , G = k 2 k 1 2 = ( p 3 p 1 ) ( p 3 p 1 ) A 2 , V = k 3 = n 2 ( v 1 ( p 3 p 1 ) ) n 1 A .
n 2 = p 2 x × p 2 y p 2 x × p 2 y | x y .
v 1 = n 2 n 1 ( v 2 ( n 2 v 2 ) n 2 ) ( 1 n 2 2 n 1 2 ( 1 ( n 2 v 2 ) 2 ) ) n 2 .
p 1 = p 2 n 0 G V Sign ( r 1 r 0 ) V 2 ( n 0 2 n 1 2 ) G v 1 .
A = Sign ( r 2 r 1 ) ( n 2 / n 0 ) ( r 2 a 2 ) + ( n 1 / n 0 ) r 1 + Sign ( r 1 r 0 ) r 0 , G = n 0 ( v 1 ( p 0 p 2 ) ) n 1 A , V = ( p 0 p 2 ) ( p 0 p 2 ) A 2 .
p 1 x × p 1 y | x y [ 0 , 0 , 0 ] ( x , y ) Aperture,
stop | | n 2 × n 2 | | d s         0 or stop ( ( n 2 × n 2 ) ( n 2 × n 2 ) ) d s 0 ,
Max { i s | n i n i | }   or Min { i s ( ( n 2 × n 2 ) ( n 2 × n 2 ) ) } ,
( p 2 p 2 ) v 1 2 = p 2 2 2 p 2 ( p 1 ( p 1 v 1 ) v 1 ) + p 1 2 + ( p 1 p 1 ) v 1 2 2 ( p 1 v 1 ) p 1 v 1 ,
2 ( p 2 p 3 ) v 1 2 = 2 ( ( p 3 v 1 ) ( p 2 p 1 ) v 1 + ( p 1 p 3 ) v 1 2 ) .
( p 2 p 2 ) v 1 2 = ( x 2 2 + y 2 2 + z 2 2 ) [ a 2 b 2 c 2 ] = [ x 2 2 a 2 + y 2 2 a 2 + z 2 2 a 2 x 2 2 b 2 + y 2 2 b 2 + z 2 2 b 2 x 2 2 c 2 + y 2 2 c 2 + z 2 2 c 2 ] .
( p 2 p 2 ) v 1 2 = [ a 2 x 2 2 + a 2 ( b ( x 2 x 1 ) a + y 1 ) 2 + a 2 ( c ( x 2 x 1 ) a + z 1 ) 2 b 2 ( a ( y 2 y 1 ) b + x 1 ) 2 + b 2 y 2 2 + b 2 ( c ( y 2 y 1 ) b + z 1 ) 2 c 2 ( a ( z 2 z 1 ) c + x 1 ) 2 + c 2 ( b ( z 2 z 1 ) c + y 1 ) 2 + c 2 z 2 2 ] , = [ a 2 x 2 2 + ( b ( x 2 x 1 ) + a y 1 ) 2 + ( c ( x 2 x 1 ) + a z 1 ) 2 ( a ( y 2 y 1 ) + b x 1 ) 2 + b 2 y 2 2 + ( c ( y 2 y 1 ) + b z 1 ) 2 ( a ( z 2 z 1 ) + c x 1 ) 2 + ( b ( z 2 z 1 ) + c y 1 ) 2 + c 2 z 2 2 ] .
( p 2 p 2 ) v 1 2 = p 2 2 + 2 p 2 [ a ( p 1 v 1 ) x 1 b ( p 1 v 1 ) y 1 c ( p 1 v 1 ) z 1 ] + p 1 2 + ( x 1 2 + y 1 2 + z 1 2 ) [ a 2 b 2 c 2 ] 2 v 1 [ a x 1 2 + b x 1 y 1 + c x 1 z 1 b y 1 2 + a x 1 y 1 + c y 1 z 1 c z 1 2 + a x 1 z 1 + b y 1 z 1 ] , = p 2 2 2 p 2 ( p 1 ( p 1 v 1 ) v 1 ) + p 1 2 + ( p 1 p 1 ) v 1 2 2 ( p 1 v 1 ) p 1 v 1 .
2 ( p 2 p 3 ) v 1 2 = 2 [ a 2 x 2 x 3 + a 2 y 2 y 3 + a 2 z 2 z 3 b 2 x 2 x 3 + b 2 y 2 y 3 + b 2 z 2 z 3 c 2 x 2 x 3 + c 2 y 2 y 3 + c 2 z 2 z 3 ] .
2 ( p 2 p 3 ) v 1 2 = 2 [ a 2 x 2 x 3 + a 2 ( b ( x 2 x 1 ) a + y 1 ) y 3 + a 2 ( c ( x 2 x 1 ) a + z 1 ) z 3 b 2 ( a ( y 2 y 1 ) b + x 1 ) x 3 + b 2 y 2 y 3 + b 2 ( c ( y 2 y 1 ) b + z 1 ) z 3 c 2 ( a ( z 2 z 1 ) c + x 1 ) x 3 + c 2 ( b ( z 2 z 1 ) c + y 1 ) y 3 + c 2 z 2 z 3 ] , = 2 [ a x 2 ( a x 3 + b y 3 + c z 3 ) b y 2 ( a x 3 + b y 3 + c z 3 ) c z 2 ( a x 3 + b y 3 + c z 3 ) ] 2 [ a 2 ( x 1 x 3 + y 1 y 3 + z 1 z 3 ) b 2 ( x 1 x 3 + y 1 y 3 + z 1 z 3 ) c 2 ( x 1 x 3 + y 1 y 3 + z 1 z 3 ) ] + 2 [ a x 1 ( a x 3 + b y 3 + c z 3 ) b y 1 ( a x 3 + b y 3 + c z 3 ) c z 1 ( a x 3 + b y 3 + c z 3 ) ] , = 2 ( p 3 v 1 ) p 2 v 1 2 ( p 1 p 3 ) v 1 2 + 2 p 1 v 1 ( p 3 v 1 ) ,
2 ( p 2 p 3 ) v 1 2 = 2 ( ( p 3 v 1 ) ( p 2 p 1 ) v 1 + ( p 1 p 3 ) v 1 2 ) .
( p 2 p 1 ) 2 2 ( v 1 ( p 3 p 1 ) ) ( p 2 p 1 ) v 1 ( p 1 ( 2 p 3 p 1 ) ) v 1 2 = ( k 1 v 1 n 1 n 2 ( p 2 p 1 ) ) 2 ( p 3 p 3 ) v 1 2 .
( p 2 p 1 ) ( p 2 p 1 2 ( v 1 ( p 3 p 1 ) ) v 1 ) = ( k 1 v 1 n 1 n 2 ( p 2 p 1 ) ) 2 ( p 3 p 1 ) ( p 3 p 1 ) v 1 2 .
k 2 = ( p 3 p 1 ) ( p 3 p 1 ) .
a 1 2 2 ( v 1 ( p 3 p 1 ) ) a 1 v 1 = ( k 1 v 1 n 1 n 2 a 1 ) 2 k 2 v 1 2 , a 1 2 2 ( v 1 ( p 3 p 1 ) ) v 1 a 1 = k 1 2 v 1 2 2 n 1 n 2 k 1 v 1 a 1 + n 1 2 n 2 2 a 1 2 k 2 v 1 2 , n 2 a 1 2 2 n 2 ( v 1 ( p 3 p 1 ) ) v 1 a 1 = n 2 k 1 2 v 1 2 2 n 1 k 1 v 1 a 1 + n 1 2 n 2 a 1 2 n 2 k 2 v 1 2 , ( n 2 2 n 1 2 n 2 ) a 1 2 + 2 ( n 1 k 1 n 2 ( v 1 ( p 3 p 1 ) ) ) v 1 a 1 = n 2 ( k 1 2 k 2 ) v 1 2 ,
( n 2 2 n 1 2 n 2 ) a 1 2 2 k 3 v 1 a 1 + n 2 ( k 2 k 1 2 ) v 1 2 = 0 ,

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