## Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces |

Optics Express, Vol. 20, Issue 10, pp. 10839-10846 (2012)

http://dx.doi.org/10.1364/OE.20.010839

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### Abstract

The two-dimensional analytic optics design method presented in a previous paper [Opt. Express **20**, 5576–5585 (2012)] is extended in this work to the three-dimensional case, enabling the coupling of three ray sets with two free-form lens surfaces. Fermat’s principle is used to deduce additional sets of functional differential equations which make it possible to calculate the lens surfaces. Ray tracing simulations demonstrate the excellent imaging performance of the resulting free-form lenses described by more than 100 coefficients.

© 2012 OSA

## 1. Introduction

1. F. Duerr, P. Benítez, J.C. Miñano, 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] [PubMed]

*θ*will have perfect focusing [2

2. 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]

*y*-coordinate dependency, Fermat’s principle is used in Sec. 2 to deduce a system of functional differential equations in two variables by using the previously established concept of convergence points [1

1. F. Duerr, P. Benítez, J.C. Miñano, 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] [PubMed]

4. D. Grabovičkić, P. Benítez, and J.C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express **19**, A747–A756 (2011). [CrossRef]

## 2. Analytic free-form solution starting from convergence points

1. F. Duerr, P. Benítez, J.C. Miñano, 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] [PubMed]

*m*

_{0}and

*m*

_{1}at the convergence points. This basic construction for one on-axis and one off-axis ray set will serve as the starting point for the three-dimensional problem as well. Figure 2(a) shows the convergence points construction in analogy to the two-dimensional problem.

*x*

_{0},

*y*

_{0},

*z*

_{0}) with

*y*

_{0}= 0 due to the overall lens’ mirror symmetry with respect to the

*x*–

*z*- and

*y*–

*z*- plane. Therefore, the normal vector at the first convergence point is completely described by the single variable

*m*

_{0}. The intersection of the refracted on-axis ray through (

*x*

_{0}, 0,

*z*

_{0}) and the refracted off-axis ray through the mirrored convergence point (−

*x*

_{0}, 0,

*z*

_{0}) determines the coordinates (

*x*

_{1}, 0,

*z*

_{1}) of the second convergence point. The normal vector at the second convergence point is also described by a the single variable

*m*

_{1}. Further refractions at the second convergence point result in the focus positions (0, 0,

*d*) and (

*r*, 0,

*d*). The initial ray construction lies within the sectional plane

*y*= 0 of the lens and corresponds exactly to the two-dimensional construction.

*z*=

*f*(

*x,y*) and

*z*=

*g*(

*x,y*) are defined to describe the lens surfaces. To completely describe the optical paths of rays passing through the lens surfaces, it is necessary to introduce additional mapping functions. In two dimensions, two mapping functions

*s*(

*x*) and

*u*(

*x*) were sufficient to define the mapping in

*x*-coordinate. For the full three-dimensional problem it is necessary to define two additional mapping functions in

*y*-coordinate, for on- and off-axis rays, respectively. This results in the overall mapping functions

*s*(

*x,y*) and

*t*(

*x,y*) for on-, and

*u*(

*x,y*) and

*v*(

*x,y*) for off-axis rays, and for

*x*-direction and

*y*-direction, respectively.

*p⃗*

_{1}= (

*x*,

*y*,

*f*(

*x,y*)) on the upper lens surface which is then refracted in

*p⃗*

_{2}= (

*s*(

*x, y*),

*t*(

*x,y*),

*g*(

*s*(

*x,y*),

*t*(

*x,y*))) towards the focal point

*p⃗*

_{3}= (0, 0,

*d*). Figure 2(c) shows an off-axis ray passing through an arbitrary point

*v⃗*

_{0}and

*v⃗*

_{1}denote the directional vectors for on- and off-axis ray sets, respectively. The position vector

*w⃗*

_{0}denotes an arbitrary but fixed point on both plane wave-fronts and

*n*

_{2}denotes the refractive index of the lens.

*y*-coordinate dependency. Therefore, additional conditional equations are required and defined as for on-axis, and as for off-axis rays, using identical arguments as in [1

**20**, 5576–5585 (2012). [CrossRef] [PubMed]

*f*(

*x,y*),

*g*(

*x,y*),

*s*(

*x,y*),

*t*(

*x,y*),

*u*(

*x,y*) and

*v*(

*x,y*). To our knowledge, the existence and uniqueness of solutions to similar systems of functional differential equations in two variables have not been discussed in detail nor proven up to now.

*f,g,s,t,u,v*) to the functional differential Eqs. (3) and (4), Taylors theorem implies that the functions must be infinitely differentiable and have a power-series representation in two variables. Thus the six functions can be given by power-series centered at convergence points (

*x*

_{0}, 0,

*z*

_{0}) and (

*x*

_{1}, 0,

*z*

_{1}), respectively. The exponents for

*y*-coordinate take already into account that all functions are either even (provided by 2

*j*) or odd (provided by 2

*j*– 1) in

*y*-direction.This symmetry follows immediately from the later introduced Eqs. (9) and (10). The linear systems of equations for the correspondent Taylor series coefficients have null vectors as only possible solutions and are therefore discarded already in the Taylor series. The coefficients

*f*

_{i}_{,0},

*g*

_{i}_{,0},

*s*

_{i}_{,0}and

*u*

_{i}_{,0}can be identified as the already calculated Taylor series coefficients with the two-dimensional optics design method. The in Fig. 2(a) introduced and in Eqs. (5)–(7) assigned initial conditions

*D*= 0 for

_{i}*i*= 1..6 and provide general solutions for the initial Taylor series coefficients depending upon variables

*m*

_{0}and

*m*

_{1}. The overall solution for arbitrary order can be calculated by solving four equations for

*y*-coordinate dependency, and six equations

*x*–

*y*-coordinate dependency. Provided that the solution for the two-dimensional problem is known, there are again two cases needed to be solved:

- For
*m*= 1, the set of Eq. (9) results in four nonlinear algebraic equations for Taylor series coefficients*f*_{0,2},*g*_{0,2},*t*_{0,1}and*v*_{0,1}. These equations have two general solutions, where one solution can be discarded as non-physical. The remaining unique solution can be expressed as functions of the initial, already known Taylor coefficients. - For
*m*≥ 1 and*n*≥ 1, it is useful to introduce an ordinal number*o*=*m*+*n*. The sets of Eqs. (9) and (10) result in two systems of linear equations for particular Taylor series coefficients which can be expressed as compact matrix equations for*y*-coordinate dependency, and for*x*–*y*-coordinate dependency and for arbitrary*o*≥ 2. The matrix elements of*M*and_{y}*M*consist of mathematical expressions which depend on Taylor series coefficients calculated for the initial conditions. The needed vector elements of_{xy}*b⃗*^{(}^{n,m}^{)}for*o*=*n*+*m*are mathematical expressions only dependent on previously calculated Taylor series coefficients for*o*= 2,3,..,(*n*+*m*–1) and can be calculated for each ordinal number*o*= 2,3,.. in ascending order and for all possible combinations of*n*and*m*from Eqs. (9) and (10). For known matrices*M*and_{y}*M*and vectors_{xy}*b⃗*^{(}^{n,m}^{)}, the Taylor series coefficients can be calculated by solving the linear systems of Eqs. (11) and (12).

*y*-coordinate dependency (

*j*≥ 1) up to an arbitrary but finite order. Taylors remainder theorem in two variables provides quantitative estimates on convergence and the approximation error of the functions by its Taylor polynomials. The radii of convergence for the expansions

*f*(

*x,y*) and

*g*(

*x,y*) are very important, as they indicate the maximum aperture that can be achieved for a given set of initial values. In the examples considered, the radius of convergence is larger than the range of 0 <

*x*<

*x*and 0 <

_{max}*y*<

*y*of the lenses. The presented calculation is fully implemented in Wolfram

_{max}*Mathematica*.

## 3. Ray tracing results for calculated high-order Taylor polynomials of two variables

*x*-coordinate dependency in [1

**20**, 5576–5585 (2012). [CrossRef] [PubMed]

*f*(

*x,y*) and

*g*(

*x,y*) and the mapping functions

*s*(

*x,y*),

*t*(

*x,y*),

*u*(

*x,y*) and

*v*(

*x,y*) for input parameters (

*θ*,

*m*

_{0},

*m*

_{1},

*n*

_{2}). For

*x*-coordinate dependency, the coefficients were previously calculated up to 15

^{th}order.

*y*-coordinate dependency are calculated up to

*o*= 13, resulting in a total number of 65 Taylor coefficients for

*f*(

*x,y*) and

*g*(

*x,y*), respectively. This derived general three-dimensional calculation provides solutions for any given (physically meaningful) initial parameter set (

*θ*,

*m*

_{0},

*m*

_{1},

*n*

_{2}).

*m*

_{0}and

*m*

_{1}for fixed design angle

*θ*= 12° and refractive index

*n*

_{2}= 1.5 with the two-dimensional optics design method, ensuring that the boundary condition at the optical axis is fulfilled. For each pair of variates, the full three-dimensional solution is then determined. The present quadrant symmetry of the calculated lens surfaces is not further imposed; it rather follows naturally from the calculated solutions. We cannot provide a mathematical proof but only state that the so far calculated free-form surfaces are always quadrant symmetric, as long as the boundary condition of the 2D solution

*f*′(0) =

*g*′(0) = 0 is fulfilled.

*m*

_{0}and

*m*

_{1}in three dimensions. The slope value

*m*

_{1}ranges from −0.065 to 0.045. For higher values, all Taylor polynomial coefficients with a

*y*-coordinate component become imaginary and the three-dimensional lens construction fails. Figure 3 shows two single-frame excerpts from this ray tracing animation video for

*m*

_{1}= −0.065 which is equivalent to a meniscus lens (a), and

*m*

_{1}= 0.045 which is equivalent to a biconvex lens (b). The two arrows indicate the directions of the normal vectors at the convergence points.

*f*(

*x,y*) and

*g*(

*x,y*) are approximated by tensor product B-spline surfaces of 3

^{rd}order defined over an

*x*–

*y*-grid. To avoid occurring approximation errors in the future, an extension of OPS or the use of a commercial ray tracer will be considered. To evaluate the imaging performance in three dimensions with OPS, the RMS spot radius is analyzed as a function of the incident direction in

*x*–

*z*-plane and for monochromatic light (corresponding to

*n*

_{2}= 1.5) with the object points at infinity (i.e., parallel rays). The ray tracing results of Fig. 4(a) and 4(b) show the RMS spot radius plotted against the incident angle in 0.5° steps for two different pairs of variates

*m*

_{0}and

*m*

_{1}and the corresponding rotational symmetric and free-form lenses.

*m*

_{1}= −0.065 for the calculated free-form lens as shown in Fig. 3(a), and for the rotational symmetric equivalent based on the two-dimensional design method as shown in Fig. 1(b). Figure 4(b) shows the same evaluation for

*m*

_{1}= 0.045. Both rotational symmetric lenses have an entrance diameter of 3.8mm; the upper surfaces of the free-form lenses have dimensions of 3.8mm×2.92mm, ensuring approximately identical entrance aperture surface areas. The f-numbers of the rotational symmetric lenses are f/2.7 for the meniscus, and f/1.9 for the biconvex lens. The free-form lenses have identical f-numbers in x-direction for the meniscus and the biconvex lens, respectively. In y-direction, the f-numbers are f/3.4 for the meniscus, and f/3.2 for the biconvex free-form lens. The four mapping functions

*s*(

*x,y*),

*t*(

*x,y*),

*u*(

*x,y*) and

*v*(

*x,y*) enable a precise tailoring of the lower lens surfaces, ensuring that all rays going through the first lens surface also reach the receiver plane.

## 4. Conclusion

## Acknowledgments

## References and links

1. | F. Duerr, P. Benítez, J.C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express |

2. | 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 |

3. | P. Benítez, J.C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. |

4. | D. Grabovičkić, P. Benítez, and J.C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

**History**

Original Manuscript: March 7, 2012

Revised Manuscript: April 13, 2012

Manuscript Accepted: April 18, 2012

Published: April 25, 2012

**Citation**

Fabian Duerr, Pablo Benítez, Juan C. Miñano, Youri Meuret, and Hugo Thienpont, "Analytic free-form lens design in 3D: coupling three ray sets using two lens surfaces," Opt. Express **20**, 10839-10846 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-10-10839

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### References

- F. Duerr, P. Benítez, J.C. Miñano, Y. Meuret, and H. Thienpont, “Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles,” Opt. Express20, 5576–5585 (2012). [CrossRef] [PubMed]
- J.C. Miñano, P. Benítez, W. Lin, J. Infante, F. Muñoz, and A. Santamaría, “An application of the SMS method for imaging designs,” Opt. Express17, 24036–24044 (2009). [CrossRef]
- P. Benítez, J.C. Miñano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng.43, 1489 (2004). [CrossRef]
- D. Grabovičkić, P. Benítez, and J.C. Miñano, “Free-form V-groove reflector design with the SMS method in three dimensions,” Opt. Express19, A747–A756 (2011). [CrossRef]

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