## Abstract

An imaging design approach for optical systems consisting of two aspheres which is free of astigmatism is presented in this paper. A set of implicit differential equations is derived from generalized ray tracing. The solution of the derived equations provides the profiles of the two aspheres as well as the object to image mapping. The obtained design can be used as a good starting point for optimization. Particular examples are given.

© 2015 Optical Society of America

## 1. Introduction

Aspheric surfaces can in general provide more compact imaging designs with better performance than their spherical counterparts. We call aspheric surface or asphere to a rotational symmetric surface which is not a portion of a sphere.

Schmidt corrector plate is an early example of a single aspheric design to correct spherical aberration in a telescope [1], and Schwarzschild pioneered the design of two aspheres for an aplanatic system [2]. Later, Wassermann and Wolf [3] generalized Schwarzschild approach presenting a method for the design of two aspheres as solution of two simultaneous first-order ordinary differential equations. Standard integrating methods have been applied to obtain the aspheric profiles numerically. More recently Willstrop and Lynden-Bell gave general analytic solutions of the two-mirror aplanatic design problem [4, 5]. The 2D-SMS (Two-dimensional Simultaneous Multiple Surfaces) method, which can be seen as a system of functional differential equations, is another design procedure that has been used to design directly two or more aspheres for imaging [6]. It has been used, for instance, to design a telephoto for the SWIR band [7]. In this design the 6 SMS aspheres fitted with Forbes polynomials [8, 9] were used as the starting point of an optimization procedure. As yet there are 2D-SMS designs perfectly imaging up to 8 fields. This method can also be used to image several fields using less aspheres, only by controlling the size of the pupil aperture [10].

Novel direct methods for designing an anastigmatic (free of astigmatism) single-asphere and an anastigmatic single-freeform-surface optical systems have been recently presented [11, 12]. These methods are based on finding a differential equation on the optical surface deduced by equating the principal curvatures of the output wavefronts. In this design procedure, the mapping between object and image points cannot be prescribed. This mapping and the shape of the image surface are obtained as a result of the design process. Here, we have extended this strategy to the design of a double asphere anastigmatic optical system. The second surface provides additional freedom and, unlike the preceding single surface method, now both the object and the image surface shapes can be prescribed. The derivation of the differential equations is done in section 3. In section 4, an anastigmatic design is presented and compared with an aplanatic design as a starting point for multi-parameter optimization.

## 2. Statement of the problem

The design method introduced here adopts the definition of “supporting” and “supported” wavefronts from [11]. The input supporting wavefront is normal to a set of rays in the object space, which once refracted or reflected by the two unknown aspheres, define the output supporting wavefront in the image space. There is a supported wavefront per each ray of the supporting wavefront. This ray is also normal to its corresponding supported wavefront. Then, the supported wavefronts are tangent to the supporting wavefront, but with different principal curvatures at the points of tangency. In our approach, we will consider a second order approximation of the supported wavefronts in the neighborhood of the point tangent to the supporting wavefront, therefore, each supported wavefront surface will be fully characterized by the tangent point, the ray direction and the value of the principal curvatures relative to those of the supporting wavefront, all these values taken at the tangent point. The design problem for double surface system is to find the optical surfaces (refractive or reflective) such that the second order approximations of the output supported wavefronts when they get the image are spheres centered at the image points, so the design is free from astigmatism. The image surface, where the centers of curvature of the spherical output supported wavefronts lay on, will be prescribed, as well as the input supporting and supported wavefronts. This description is adequate to model an optical system with a small aperture stop. A schematic drawing of the system is shown in Fig. 1.

These design conditions lead to a group of three implicit ordinary differential equations in the general case as shown in the next section.

## 3. Differential equations for double optical surface profiles

We consider a general double surface optical system with refractive index *n*_{2} between two surfaces and *n*_{1} elsewhere shown in Fig. 2. We will restrict our analysis henceforth to the particular case in which the input supporting wavefront is a sphere centered at the origin **O**.

(*r _{f} (α)*,

*α*) and (

*r*,

_{s}(β)*β*) are points on the first and the second surface respectively, in two polar coordinates with the same origin

**O**.

*θ*

_{1},

*θ*

_{2},

*θ*

_{3}and

*θ*

_{4}are angles between the ray vector and the surface normal. The prescribed image surface is given in implicit form by

*S(x, y)*= 0. The two surface profiles are defined by the three unknown functions

*r*,

_{f}(α)*r*and

_{s}(β)*β(α)*.

According to the generalized ray tracing equations [13], the change of the curvatures of the supported wavefronts due to the refractions are expressed by the following equations:

*q*and

*p*refer respectively to the incidence plane and its perpendicular plane (both contain the normal to the surface);

*ρ*together with the suffix

*f*and

*s*refers to the radii of curvature of the first and the second optical surface respectively;

*ρ*together with the suffix 1 and 2 refers to the radii of curvature of the wavefronts before and after the first refraction;

*ρ*together with the suffix 3 and 4 refers to the radii of curvature of the wavefronts before and after the second refraction. The sign convention for the radius of curvature in this paper follows the one stated by Smith [14].

The radii of curvature of the wavefronts between refractions are expressed by:

*D*is the distance between the two points of refraction:

Anastigmatic designs require *ρ _{q}*

_{4}=

*ρ*

_{p}_{4}. Then the supported wavefronts are locally spherical after the second refraction. Henceforth we shall restrict the analysis to the case where the second order approximation of the input supported wavefronts are spheres with radius

*R*(

*α*) at the point of the first refraction. So

*ρ*

_{q}_{4}=

*ρ*

_{p}_{4}) we can eliminate the 8 variables

*ρ*,

_{qi}*ρ*(

_{pi}*i*= 1, 2, 3, 4) to get a single equation:

The loci of the curvature centers of the output supported wavefronts are calculated from the radii of curvature and the direction of the output supported wavefronts. Since the curvature centers have to be on the prescribed image surface, the output supported wavefronts and the prescribed image surface are connected by:

Substituting these equations into *S(x, y)* = 0, we get

Now get the expression of *ρ _{p}*

_{4}from the second equations in Eqs. (1)-(3) as

Finally, let’s derive the following equation from the sine law applied to the triangle formed by the origin and the two points on the two surfaces where the refractions occur:

Consider now Eq. (6) after substituting *D* (Eq. (4)) into it; Eq. (8) after substituting *ρ _{p}*

_{4}(Eq. (9)) and

*D*(Eq. (4)) into it; and Eq. (10). These three equations relate the variables

*n*

_{1},

*n*

_{2},

*R*,

*α*,

*β*,

*r*,

_{f}*r*,

_{s}*ρ*,

_{pf}*ρ*,

_{qf}*ρ*,

_{ps}*ρ*,

_{qs}*θ*

_{1},

*θ*

_{2},

*θ*

_{3},

*θ*

_{4}and the function

*S(x, y)*= 0.

*ρ*,

_{pf}*ρ*,

_{qf}*ρ*and

_{ps}*ρ*can be expressed as functions of

_{qs}*α*,

*β*, and

*r*´,

_{f}*r*´´,

_{f}*r*´,

_{s}*β*´ (where ′ = d/d

*α*and ″ = d

^{2}/d

*α*

^{2}) using the general formulas for rotational surfaces [15] as:

*θ*

_{1}and

*θ*

_{3}can also be written as functions of

*α*,

*β*,

*r*,

_{f}*r*and

_{s}*β*´,

*r*´,

_{f}*r*´

_{s}*θ*

_{2}and

*θ*

_{4}can be obtained by Snell’s law from

*θ*

_{1}and

*θ*

_{3}. Then

*θ*

_{1},

*θ*

_{2},

*θ*

_{3}and

*θ*

_{4}can also be written as functions of

*α*,

*β*,

*r*,

_{f}*r*and

_{s}*β*´,

*r*´,

_{f}*r*´.

_{s}Since *n*_{1}, *n*_{2}, *R(α)* and the function *S(x, y)* = 0 is known, the three equations Eqs. (6), (8) and (10) are in fact three implicit ordinary differential equations with three unknown functions *r _{f}(α)*,

*r*and

_{s}(β)*β(α)*. The numerical solutions of these functions represent aspheric profiles.

## 4. Optical design example and evaluation

A design example has been derived using the approach presented in section 3.

For this example *R*(*α*) = ∞. The vertexes of the first and the second optical surface are on the axis of the rotational symmetry, 31.5 mm and 41.5 mm away from the origin respectively; the image surface is a flat plane perpendicular to the optical axis and is 85 mm away from the origin (*S*(*x*, *y*)≡*x*-85mm = 0), so that Eq. (8) becomes

*n*

_{1}= 1,

*n*

_{2}= 1.49 and the lens diameter is 42.4 mm. The design can be obtained by solving the three implicit ordinary differential equations using the ode15i function of Matlab. The function requires values of unknown functions and their derivatives at the initial point as boundary conditions. In this example, they are

*r*,

_{f}*r*´,

_{f}*r*´´,

_{f}*β*,

*β*´,

*r*,

_{s}*r*´/

_{s}*β*´ and (

*r*´/

_{s}*β*´)´ at

*α*=

*α*

_{0}. To avoid singular point on the symmetrical axis,

*α*

_{0}is chosen as a small increment from 0° so as to maintain the prescribed surface positions. Thus

*r*= 31.5,

_{f}*r*= 41.5. When

_{s}*r*´ is chosen,

_{f}*β*can be obtained by substituting

*r*,

_{f}*r*and

_{s}*r*´ into Eq. (10). Then

_{f}*r*´/

_{s}*β*´ can be obtained by solving Eq. (8). Finally

*r*´´,

_{f}*β*and (

*r*´/

_{s}*β*´)´ can be obtained by solving Eq. (6), differentials of Eqs. (8) and (10) simultaneously. As we can see, the initial value of

*r*´ will affect the boundary condition which leads to different numerical solutions. A series value of

_{f}*r*´ has been tested and a design in which the two surfaces converge to the same edge without total internal reflection is obtained and shown in Fig. 3(a).

_{f}Once the design has been done, the two image surfaces where the centers of curvature of the tangential and sagittal output supported wavefronts lay on respectively are calculated separately from the surface profiles to check that the design is free from astigmatism. The difference between the two image surfaces and the image plane *x* = 85 mm is less than 1 μm.

An aplanatic design with the same positions of lens surface vertexes and image plane (Fig. 3(b)) has been used as comparison. Since this design is not anastigmatic there will be in general two surfaces where the centers of curvature of output wavefronts lie and these two surfaces are in general curved unlike the flat single image plane of the anastigmatic design.

All the optical surfaces are fitted and represented by Qcon asphere polynomials [8] in CodeV. A subsequent optimization has been done with a 2mm diameter round pupil centered at the origin in CodeV for 25 field points uniformly distributed in the interval 0° ≤ *α* ≤ 29.6°. The layouts of the both designs after optimization are plotted in Fig. 3(c) and (d) respectively.

The optimization and the following evaluation have been done for a monochromatic light of wavelength 580 nm. The RMS spot diameter distribution is shown in Fig. 4. The result optimized from anastigmatic design yields an average RMS spot diameter of 9.9 μm and a maximum value of 12.9 μm across 25 field points, while the result optimized from aplanatic design yields an average value of 13 μm and a maximum value of 19 μm.

The Modulation Transfer Function has been evaluated across all the 25 field points and the result is shown in Fig. 4 for both optimized designs. The result optimized from anastigmatic design has achieved an average MTF of 46.9% and a minimum of 42.8% at the frequency of 30 cycles/mm across 25 field points, while the result optimized from aplanatic design achieves an average MTF of 44.5% and a minimum of 32.3% at the same frequency.

## 5. Discussion and conclusion

The differential equation design approach shown hereinabove can control the tangential and sagittal ray propagation simultaneously, thus anastigmatic design can be readily obtained. We have extended this approach from single optical surface designs to double aspheric surface designs. With more freedoms provided by the second optical surface, both the object and the image surfaces can be prescribed. Nevertheless, in no case, nor single nor double surface design, the mapping from the object to image can be prescribed. This mapping is obtained after the design of the optical surfaces.

Based on this approach, a double aspheric surface anastigmatic design example has been developed. As a good initial design for optimization, a result from the final design with an average RMS spot diameter of 9.9 μm and an average MTF value of 46.9% at the frequency of 30 cycs/mm has been achieved.

## Acknowledgment

Authors thank the European Commission (SMETHODS: FP7-ICT-2009-7 Grant Agreement No. 288526, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (ENGINEERING METAMATERIALS: CSD2008-00066, SEM: TSI-020302-2010-65 SUPERRESOLUCION: TEC2011-24019, SIGMAMODULOS: IPT-2011-1441-920000, PMEL: IPT-2011-1212-920000), UPM (Q090935C59) and the academic licence for CodeV from Synopsys for the support given to the research activity of the UPM-Optics Engineering Group, making the present work possible.

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