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

Optics Express, Vol. 20, Issue 5, pp. 5576-5585 (2012)

http://dx.doi.org/10.1364/OE.20.005576

Acrobat PDF (1080 KB)

### 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.

© 2012 OSA

## 1. Introduction

*A*and

*B*through media with refractive index distribution

*n*. As done in most textbooks on optics, it can be used to describe the properties of light rays refracted through different media, reflected off mirrors or undergoing total internal reflection. Indeed, all known laws of geometrical optics, lens design and aberrations will be consequences of the analytic properties of solutions to Fermat’s principle [1]. However, its mathematical complexity could give rise to the impression that it is impractical to be used for optics design. In fact, conventional optics design is based on minimizing a chosen merit function which quantifies the system’s performance for a defined sets of rays. In case of imaging applications, this can be merit functions such as the sum of the squares of certain aberrations or the RMS blur spot at the image plane [2]. In case of nonimaging optics, different merit functions such as the contrast ratio at a receiver plane or the optical collection efficiency could be chosen.

3. J. Chaves, *Introduction to Nonimaging Optics* (CRC, 2008). [CrossRef]

*N*optical surfaces can couple

*N*sets of rays for which specific conditions are imposed.

4. 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]

^{th}order.

## 2. Initial degrees of freedom for SMS2D designs

*N*optical surfaces using

*N*one-parameter ray sets for which specific conditions connect the initial with the final ray sets [3

3. J. Chaves, *Introduction to Nonimaging Optics* (CRC, 2008). [CrossRef]

*N*ray sets at the correspondent

*N*image points. The SMS2D method offers the flexibility to choose the ray sets and their associated image points [5

5. 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]

*θ*= ±5° design angle and increasing lens thicknesses from left to right. Similarly, Fig. 1(d)–(f) shows designs with increasing lens thicknesses for

*θ*= ±10° design angle.

*SMS skinning*process can finally be used to fill the gaps between the separated points, fully defining both optical surfaces. A segment can be interpolated between the two symmetric adjacent points of the bottom surface, as indicated as solid lens profile in Fig. 1. As a boundary condition, the segment has to match point coordinates and normal vectors at the initial points, which requires at least a 2

^{nd}order polynomial function. However, in principle, any analytical function satisfying this boundary condition can be applied - which correspondents to an infinite parameter space for the initial segment problem. For small design angles and moderate lens thicknesses, the initial segment represents only a small fraction of the entire lens profiles. However, with increasing design angles and/or distances between the optical surfaces the relative importance of the initial segment increases as well. In extreme cases, the initial segment can completely define the full lens profiles as shown in Fig. 1(f). A selected 2

^{nd}order polynomial segment may guarantee coupling of two ray sets with chosen design angles of opposite sign, but it does not make any further use of the full potential offered by an unrestricted initial segment satisfying the boundary condition. Therefore, the main objective of this work is to find ways to construct initial segments which ensure maximum benefit from these infinite degrees of freedom.

## 3. Optimum utilization of the initial segment

_{1}and Ŝ

_{2}of the two surfaces shown in Fig. 2(a). They are designed to couple a tilted parallel ray set onto a point. Because of the symmetry applied with respect to the optical axis of the lens, the lens segments Ŝ

_{1}and S

_{2}will couple the parallel ray set with opposite sign. That means that only one ray set is actually coupled using these two surface segments. This offers the opportunity to couple an additional ray set. One of these two identical off-axis ray sets is now replaced in Fig. 2(b) by an on-axis ray set. The constant optical path length condition can then be used to calculate the final lens portion on the second lens profile by tracing off-axis rays through Ŝ

_{1}, which is illustrated in Fig. 2(c). Finally, the full set of rays on the completed lens is shown in Fig. 2(d) - coupling three ray sets with two lens profiles.

_{3}and its correspondent normal vector are mirrored along the optical axis resulting in Q

_{3}and its associated normal vector. An on-axis ray is refracted at Q

_{3}, P

_{4}is calculated using the constant optical path length condition. The correspondent normal vector is calculated to refract the ray towards the focus position R

_{1}. A ray coming from R

_{2}is refracted at P

_{4}, P

_{5}is calculated using the constant optical path length condition, as shown in Fig. 3(d). Figure 3(e) shows the next step in analogy to (c). These two steps in (c) and (d) are now alternately repeated until a stop criterion is reached.

## 4. Analytic solution of initial segments starting from convergence points

*x*

_{0},

*z*

_{0}) of the convergence point on the first lens profile can be freely chosen without loss of generality. The slope

*m*

_{0}at the convergence point represents a first variable. The intersection of the refracted on-axis ray through (

*x*

_{0},

*z*

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

*x*

_{0},

*z*

_{0}) determines the coordinates (

*x*

_{1},

*z*

_{1}) of the convergence point on the 2

^{nd}lens profile. The slope

*m*

_{1}at this second convergence point represents a second variable. The intersection of the refracted on-axis ray through (

*x*

_{1},

*z*

_{1}) and the optical axis determines the first focus position

*d*. Finally, the refracted off-axis ray and the known lateral detector position determine the x-coordinate

*r*of the second focus position. This formulation provides a compact representation of all initial values depending upon the two variables

*m*

_{0}and

*m*

_{1}only. In addition, it provides the permissible range of values for these two variables for which the convergence point construction is possible. For example, a given value

*m*

_{0}limits the possible range of values of

*m*

_{1}for which an on-axis ray can be refracted towards the optical axis.

*f*(

*x*) and

*g*(

*x*) are introduced to describe the two lens profiles. To analytically describe the optical paths of rays passing through the lens it is necessary to introduce two additional mapping functions

*s*(

*x*) and

*u*(

*x*). Figure 4(b) shows an on-axis ray passing through an arbitrary point

*p⃗*= (

*x*,

*f*(

*x*)) on the first lens profile which is then refracted in (

*s*(

*x*),

*g*(

*s*(

*x*))) towards the focal point (0,

*d*). The auxiliary function

*s*(

*x*) thus defines the mapping in x-coordinate. Similarly, function

*u*(

*x*) defines the mapping in x-coordinate for off-axis rays through an arbitrary point

*u*(

*x*),

*g*(

*u*(

*x*))) on the second lens profile, as shown in Fig. 4(c). All optical path lengths can then be expressed in sections using vector geometry as for on-axis rays, and as for off-axis rays. The vectors

*n⃗*

_{0}and

*n⃗*

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

*w⃗*

_{0}denotes a fixed point on both plane wave fronts and

*n*

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

*w⃗*

_{0}and

*n⃗*

_{0}, and the fixed point (

*s*(

*x*),

*f*(

*s*(

*x*)) on the second lens profile: an on-axis ray coming from the wave front and passing through (

*s*(

*x*),

*f*(

*s*(

*x*)) must be such that the combined optical path length

*d*

_{1}+

*d*

_{2}is an extremum. With point (

*s*(

*x*),

*f*(

*s*(

*x*)) kept fixed, the only remaining variable to achieve an extremum for

*D*

_{1}=

*d*

_{1}+

*d*

_{2}is the point (

*x*,

*f*(

*x*)) on the upper lens profile. Fermat’s principle thus implies that where the partial derivative indicates that (

*s,g*(

*s*)) is kept fixed. Similarly, an on-axis ray between fixed points (

*x*,

*f*(

*x*)) and (0,

*d*) must satisfy equation These two functional differential equations arising from Fermat’s principle describe all on-axis ray paths through the lens profiles

*f*(

*x*) and

*g*(

*x*). In analogy,

*f*(

*x*) and

*g*(

*x*) must also satisfy the functional differential equations for off-axis rays, using the same arguments as before. The lens design shown in Fig. 4 is thus fully described by four functional differential equations (Eqs. (4)–(7)) for four unknown functions

*f*(

*x*),

*g*(

*x*),

*s*(

*x*) and

*u*(

*x*). The fundamental analysis of a similar system of functional differential equations has been discussed [6

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

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

*f,g,s,u*) is an analytic and smooth solution to the functional differential equations (4)–(7), Taylor’s theorem implies that the functions must be infinitely differentiable and have a power-series representation. Thus the four functions can be given by power-series centered at convergence points (

*x*

_{0},

*z*

_{0}) and (

*x*

_{1},

*z*

_{1}), respectively. The initial conditions as introduced in Fig. 4(a) then satisfy the conditional equations

*D*= 0 for

_{i}*i*= 1..4 and provide general solutions for the initial Taylor coefficients

*f*

_{0},

*f*

_{1},

*g*

_{0},

*g*

_{1},

*s*

_{0}and

*u*

_{0}depending upon variables

*m*

_{0}and

*m*

_{1}. In ascending order it is now possible to calculate (n+1)

^{th}order Taylor series coefficients in

*f*(

*x*) and

*g*(

*x*) and n

^{th}order in

*s*(

*x*) and

*u*(

*x*) by solving equations The case for

*n*= 0 corresponds to the just solved equations for initial Taylor coefficients. There are two further cases needed to be solved:

- For
*n*= 1, the set of Eq. (11) results in nonlinear algebraic equations for Taylor series coefficients*f*_{2},*g*_{2},*s*_{1}and*u*_{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*n*= 0. - For
*n*> 1, the set of Eq. (11) results in linear algebraic equations for particular Taylor series coefficients. By sorting and combining terms, the equations (11) can be transformed and expressed as a compact matrix equation for arbitrary*n*> 1. The matrix elements*M*consist of mathematical expressions which depend on Taylor series coefficients obtained for_{ij}*n*= 0,1. The vector elements of*b⃗*^{(n)}on the right hand side of Eq. (12) are mathematical expressions only dependent on previous Taylor series coefficients*f*,_{i}*g*,_{i}*s*_{i}_{−1}and*u*_{i−1}for*i*= 0..*n*. Finally, the vector elements of*b⃗*^{(n)}can be calculated for each*n*= 2,3,4,.. (in this order) from Eq. (11). For known matrix*M*and vectors*b⃗*^{(2)}..*b⃗*^{(}^{n}^{)}, the Taylor series coefficients*f*_{n+1},*g*_{n+1},*s*and_{n}*u*can then be calculated by solving the linear system of Eq. (12)._{n}

*n*= 0 and

*n*= 1 and the introduced algebraic system of linear equations (12) allow to calculate all Taylor series coefficients of

*f*(

*x*),

*g*(

*x*),

*s*(

*x*) and

*u*(

*x*) up to an arbitrary order. However, a Taylor series is a representation of an analytic function as an infinite sum of terms. Indeed, it is only possible to calculate a finite number of initial terms of the Taylor series. Such a function is called a Taylor polynomial and will be the only approximation made. Furthermore, Taylor’s remainder theorem provides quantitative estimates on convergence and the approximation error of the function by its Taylor polynomial. The radii of convergence for the expansions

*f*(

*x*) and

*g*(

*x*) 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*of the lenses. The algebraic steps of calculation presented in the this section are fully implemented and calculated in Wolfram

_{max}*Mathematica*.

## 5. Ray tracing results for calculated 15^{th} order Taylor polynomials

*n*= 0,1 are calculated which also determines the matrix

*M*of Eq. (12). The general solution of the matrix equation

*Mx⃗*=

*b⃗*is then calculated to obtain the general solution vector

*x⃗*. The vector elements of

*b⃗*

^{(n)}are calculated within a loop from

*n*= 2 up to

*n*= 14. It is possible to calculate even higher orders. However, the error made in this approximation is already extremely small.

*f*(

*x*) and

*g*(

*x*) up to 15

^{th}order, and the mapping functions

*s*(

*x*) and

*u*(

*x*) up to 14

^{th}order as output arguments. The off-axis design angle

*θ*, the refractive index

*n*

_{2}and the derivatives at the convergence points

*m*

_{1}and

*m*

_{2}are passed as input arguments.

*θ*= 12°,

*n*

_{2}= 1.5), the only remaining free parameters to vary are

*m*

_{0}and

*m*

_{1}. Furthermore, the intended real focus of the lens suggest to choose a negative value for

*m*

_{0}. The sign of

*m*

_{1}then determines the global lens shape - which can be either meniscus like (negative

*m*

_{1}) or biconvex (positive

*m*

_{1}). Local smooth and analytic solutions exist for any values within the permissible range of values for these two variables. However, the lens’ smoothness and symmetry additionally requires

*f*′(0) =

*g*′(0) = 0 at the optical axis. This boundary condition introduces an additional correlation between

*m*

_{0}and

*m*

_{1}, meaning that for a specific value

*m*

_{1}the correspondent value

*m*

_{0}can be obtained by making sure

*f*′(0) =

*g*′(0) = 0 is fulfilled. Even though it is not possible to deduce this correlation

*m*

_{1}(

*m*

_{0}) as a closed form solution, the numerical calculation of roots of

*f*′(0) and

*g*′(0) is very accurate and fast.

*m*

_{0}and

*m*

_{1}. For

*m*

_{1}ranging from −0.065 to 0.065 the correspondent

*m*

_{0}, satisfying the boundary condition, is directly calculated each time. Figure 5 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}= 0.065 which is equivalent to a biconvex lens (b). The two arrows indicate the directions of the normal vectors at the convergence points.

*s*(

*x*) and

*u*(

*x*) are used to directly calculate the semi-diameters of the lens profiles. For example, solving equation

*u*(

*r*

_{1}) = 0 provides the semi-diameter

*r*

_{1}of the upper lens profile. Lens specific parameters such as the effective focal length and magnification can be directly calculated for each correspondent

*m*

_{0}and

*m*

_{1}.

*θ*= 12° and

*n*

_{2}= 1.5 have been selected arbitrarily to provide clear evidence that it is possible to calculate analytical high order Taylor polynomials for various values

*m*

_{0}and

*m*

_{1}. However, this calculated general solution provides much more. It solves the stated initial segment problem for any given (physically meaningful) initial parameter set (

*θ*,

*m*

_{0},

*m*

_{1},

*n*

_{2}). This also means that it could be used in a hybrid technique: first calculate a highly accurate initial segment and finish the overall lens design using the SMS2D design method. This particular convergence point design was chosen to demonstrate the possibility of coupling an additional on-axis ray set, meaning that three ray sets can be coupled with two surfaces. In addition, the imposed axial symmetry of the lens is not explicitly needed by this design method. Other wave fronts besides plane and spherical wave fronts could be coupled as well. A further important step will be a solution and implementation of this new method in three dimensions and for more than two surfaces. This would allow calculating free-form optics targeting applications with wide and even asymmetric field of view.

## 6. Conclusion

^{th}order demonstrated the capabilities and versatility of this new analytic optics design method. Future work will focus on its non-rotational symmetric three-dimensional generalization and target applications where this new design method can help to further increase the overall optical system’s performance.

## Acknowledgments

## References and links

1. | M. Born and E. Wolf, |

2. | P. Mouroulis and J. Macdonald, |

3. | J. Chaves, |

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

5. | 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 |

6. | A. Friedman and J.B. McLeod, “Optimal design of an optical lens,” Arch. Rational Mech. Anal. |

7. | A. Friedman and J.B. McLeod, “An optical lens for focusing two pairs of points,” Arch. Rational Mech. Anal. |

8. | 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. |

9. | B. van Brunt and J.R. Ockendon, “A lens focusing light at two different wavelengths,” J. Math. Anal. App. |

10. | B. van Brunt, “Mathematical possibility of certain systems in geometrical optics,” J. Opt. Soc. Am. A |

**OCIS Codes**

(080.2720) Geometric optics : Mathematical methods (general)

(080.2740) Geometric optics : Geometric optical design

**History**

Original Manuscript: December 9, 2011

Revised Manuscript: February 9, 2012

Manuscript Accepted: February 15, 2012

Published: February 22, 2012

**Citation**

Fabian Duerr, Pablo Benítez, Juan C. Miñano, Youri Meuret, and Hugo Thienpont, "Analytic design method for optimal imaging: coupling three ray sets using two free-form lens profiles," Opt. Express **20**, 5576-5585 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-5-5576

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### References

- 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. Express18, 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. Express17, 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. A11, 2905–2914 (1994). [CrossRef]

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