Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles

Fabian Duerr; Pablo Benítez; Juan C. Miñano; Youri Meuret; Hugo Thienpont

doi:10.1364/OE.20.005576

Abstract

In this work, a new two-dimensional optics design method is proposed that enables the coupling of three ray sets with two lens surfaces. The method is especially important for optical systems designed for wide field of view and with clearly separated optical surfaces. Fermat’s principle is used to deduce a set of functional differential equations fully describing the entire optical system. The presented general analytic solution makes it possible to calculate the lens profiles. Ray tracing results for calculated 15^th order Taylor polynomials describing the lens profiles demonstrate excellent imaging performance and the versatility of this new analytic design method.

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References

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M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1986).
P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, USA, 1997).
J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
[Crossref]
D. Grabovičkić, P. Benítez, and J.C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[Crossref]
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]
A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. 99, 147–164 (1987).
[Crossref]
A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. 101, 57–83 (1988).
[Crossref]
J.C.W. Rogers, “Existence, uniqueness and construction of the solution of a system of ordinary functional differential equations with applications to the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]
B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. 165, 156–179 (1992).
[Crossref]
B. van Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A 11, 2905–2914 (1994).
[Crossref]

1992 (1)

B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. 165, 156–179 (1992).
[Crossref]

1988 (2)

A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. 101, 57–83 (1988).
[Crossref]

J.C.W. Rogers, “Existence, uniqueness and construction of the solution of a system of ordinary functional differential equations with applications to the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

1987 (1)

A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. 99, 147–164 (1987).
[Crossref]

Benítez, P.

D. Grabovičkić, P. Benítez, and J.C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[Crossref]

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]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1986).

Chaves, J.

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
[Crossref]

Friedman, A.

A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. 101, 57–83 (1988).
[Crossref]

A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. 99, 147–164 (1987).
[Crossref]

Grabovickic, D.

D. Grabovičkić, P. Benítez, and J.C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[Crossref]

Infante, J.

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]

Lin, W.

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]

Macdonald, J.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, USA, 1997).

McLeod, J.B.

A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. 101, 57–83 (1988).
[Crossref]

A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. 99, 147–164 (1987).
[Crossref]

Miñano, J.C.

D. Grabovičkić, P. Benítez, and J.C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[Crossref]

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]

Mouroulis, P.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, USA, 1997).

Muñoz, F.

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]

Ockendon, J.R.

B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. 165, 156–179 (1992).
[Crossref]

Rogers, J.C.W.

J.C.W. Rogers, “Existence, uniqueness and construction of the solution of a system of ordinary functional differential equations with applications to the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

Santamaría, A.

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]

van Brunt, B.

B. van Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A 11, 2905–2914 (1994).
[Crossref]

B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. 165, 156–179 (1992).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1986).

Arch. Rational Mech. Anal. (2)

A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. 99, 147–164 (1987).
[Crossref]

A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. 101, 57–83 (1988).
[Crossref]

IMA J. Appl. Math. (1)

J.C.W. Rogers, “Existence, uniqueness and construction of the solution of a system of ordinary functional differential equations with applications to the design of a perfectly focusing symmetric lens,” IMA J. Appl. Math. 41, 105–134 (1988).
[Crossref]

J. Math. Anal. App. (1)

B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. 165, 156–179 (1992).
[Crossref]

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

B. van Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A 11, 2905–2914 (1994).
[Crossref]

Opt. Express (2)

D. Grabovičkić, P. Benítez, and J.C. Miñano, “Aspheric V-groove reflector design with the SMS method in two dimensions,” Opt. Express 18, 2515–2521 (2010).
[Crossref]

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]

Other (3)

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1986).

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford University Press, USA, 1997).

J. Chaves, Introduction to Nonimaging Optics (CRC, 2008).
[Crossref]

Supplementary Material (1)

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Fig. 1 Impact of increasing design angles and lens thicknesses on the relative importance of the initial segment

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Fig. 2 Shown steps of the design concept illustrate how to make use of symmetry about the optical axis to couple an additional on-axis ray set

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Fig. 3 Algorithmic implementation: shown figures (a)–(e) explain the design steps towards the final lens profiles shown in (f)

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Fig. 4 Introduction of all necessary initial values and functions to derive the conditional equations from Fermat’s principle

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Fig. 5 Exemplary single-frame excerpts from a ray tracing animation video ranging from meniscus lenses (a) to biconvex lenses (b) ( Media 1)

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Equations (12)

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

S = \int_{A}^{B} n d s

\begin{array}{l} d_{1} = {\vec{n}}_{0} \cdot (\vec{p} - {\vec{w}}_{0}) \\ d_{2} = n_{2} \sqrt{{(x - s (x))}^{2} + {(f (x) - g (s (x)))}^{2}} \\ d_{3} = \sqrt{s {(x)}^{2} + {(d - g (s (x)))}^{2}} \end{array}

\begin{array}{l} {\hat{d}}_{1} = {\vec{n}}_{1} \cdot (\hat{\vec{p}} - {\vec{w}}_{0}) \\ {\hat{d}}_{2} = n_{2} \sqrt{{(- x - u (x))}^{2} + {(f (x) - g (u (x)))}^{2}} \\ {\hat{d}}_{3} = \sqrt{{(r - u (x))}^{2} + {(d - g (u (x)))}^{2}} \end{array}

D_{1} = \frac{\partial}{\partial x} (d_{1} + d_{2}) = 0

D_{2} = \frac{\partial}{\partial s} (d_{2} + d_{3}) = 0.

D_{3} = \frac{\partial}{\partial x} ({\hat{d}}_{1} + {\hat{d}}_{2}) = 0

D_{4} = \frac{\partial}{\partial u} ({\hat{d}}_{2} + {\hat{d}}_{3}) = 0

\begin{matrix} f (x) = \sum_{i = 0}^{\infty} f_{i} {(x - x_{0})}^{i} & g (x) = \sum_{i = 0}^{\infty} g_{i} {(x - x_{1})}^{i} \end{matrix}

\begin{matrix} s (x) = \sum_{i = 0}^{\infty} s_{i} {(x - x_{0})}^{i} & u (x) = \sum_{i = 0}^{\infty} u_{i} {(x - x_{0})}^{i} \end{matrix}

\begin{array}{l} f (x_{0}) = z_{0} & f^{'} (x_{0}) = m_{0} \\ g (x_{1}) = z_{1} & g^{'} (x_{1}) = m_{1} \\ s (x_{0}) = x_{1} & u (x_{0}) = x_{1} \end{array}

\begin{matrix} lim_{x \to x_{0}} \frac{\partial^{n}}{\partial x^{n}} D_{i} = 0 & (i = 1..4), & {n \in 𝕅_{1}} . \end{matrix}

M (\begin{matrix} f_{n + 1} \\ g_{n + 1} \\ s_{n} \\ u_{n} \end{matrix}) = (\begin{matrix} b_{1}^{(n)} \\ b_{2}^{(n)} \\ b_{3}^{(n)} \\ b_{4}^{(n)} \end{matrix})

Abstract

References

2010 (1)

2009 (1)

1994 (1)

1992 (1)

1988 (2)

1987 (1)

Benítez, P.

Born, M.

Chaves, J.

Friedman, A.

Grabovickic, D.

Infante, J.

Lin, W.

Macdonald, J.

McLeod, J.B.

Miñano, J.C.

Mouroulis, P.

Muñoz, F.

Ockendon, J.R.

Rogers, J.C.W.

Santamaría, A.

van Brunt, B.

Wolf, E.

Arch. Rational Mech. Anal. (2)

IMA J. Appl. Math. (1)

J. Math. Anal. App. (1)

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

Opt. Express (2)

Other (3)

Supplementary Material (1)

Cited By

Figures (5)

Equations (12)

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