## Abstract

We address, in detail, the system of differential equations determining a freeform aplanatic system with illustrative examples. We also demonstrate how two optical surfaces, in general, are insufficient in achieving freeform aplanatism through the use of integrability condition for a given reflective freeform aplanatic configuration. This result also alludes to the fact that a freeform aplanatic system fulfills a broader set of conditions than its rotationally symmetric counterpart. We also elaborate on the above results with two illustrative examples (1) A semi aplanatic system which satisfies the generalized sine condition in only one direction and (2) A fully freeform aplanatic reflective system.

© 2017 Optical Society of America

## 1. Introduction

Traditional optical designs consisted, in general, of rotationally symmetric shapes owing to limitations of fabrication technologies. Advances in manufacturing of optical systems has paved way for the fabrication of optical systems having no rotational symmetry which have come to be known as freeform surfaces [1]. Introduction of freeform surfaces have aided the designers with additional degrees of freedom to ensure more control over the rays resulting in designs successful in solving optical problems having rotationally asymmetric prescription such as those in head-lamps and other situations described in [2–4]. On the other hand, optical systems that are free from spherical aberration and circular coma are known as aplanatic systems since Ernst Abbe demonstrated that adherence to the Abbe sine condition can eliminate circular coma for a microscopic objective [5]. Aplanatic systems can also be defined as optical systems which image stigmatically one point in the object space to a point in the image space and satisfy the Abbe sine condition. Aplanatism here refers to full aplanatism, i.e. the aberrations vanish for all orders. Many studies have already reported extensively about aplanatism in the last century. An excellent summary of such studies can be found in [6]. Two-mirror rotationally symmetric aplanatic systems have already been extensively studied before [7,8] and were primarily designed for telescopes which required superior aberration-free imaging characteristics. With increasing use of freeform surfaces in optical designs, it becomes significant to explore aplanatism under freeform prescription. In a previous manuscript [9], we have addressed freeform aplanatism and its link to the SMS design method. In this paper, which can be treated as a follow up to the previous publication, we give a mathematical formulation defining a three-surface freeform aplanat. Also, we formally prove that a two-surface configuration cannot, in general, satisfy the conditions required for a freeform aplanatic system.

## 2. Aplanatism in rotationally symmetrical systems

The sine condition is a mapping prescription for the rays imaging stigmatically 2 points which in general, are the origins of coordinates in object and image spaces. Systems failing to adhere to sine condition suffer from aberrations which have a linear dependency over field [10]. This sine condition establishes that *p* = *M p*´ for the rays emanating from the origin of the object plane, which must be stigmatically imaged on the origin of the image plane. *p* and *p*´ are the optical direction cosines of the ray with respect to the *x* axis at the object and at the image space respectively. *M* is a constant called magnification. Because of the rotational symmetry the same equation holds for *q* and *q*´ (direction cosines with respect to *y* axis), i.e., *q* = *M q*´. The variables *p*, *q*, and *p*´, *q*´ are the optical direction cosines with respect to the *x*, *y* axes (object plane) and the *x*´, *y*´ axes (image plane).

## 3. Aplanatism in freeform optical systems – How many surfaces does it take?

In an earlier manuscript, we have already established the link between SMS (Simultaneous Multiple Surfaces) design method and freeform aplanatic systems [9] and using that perspective we proved that in general two optical surfaces do not provide enough degrees of freedom to satisfy the aplanatism condition in freeform problems. In this section, we show how two surfaces are insufficient from a perspective that does not involve the relationship between the SMS method and aplanatic systems.

The Abbe sine condition generalized to the freeform case establishes that the rays stigmatically imaging the two origins must also fulfill *p*´ = *p*_{0}´ + *p/ M*_{X}, *q*´ = *q*_{0}´ + *q/ M*_{Y}, where *p*_{0}, *q*_{0}, *M*_{X}, and *M*_{Y} are constants (the last two are called magnifications, see [Eq. (7)] in [9]). This mapping prescription appears as a necessary condition when requiring imaging properties for a point at the object and its neighborhood. These types of constraints were analyzed by Stone and Forbes for a general asymmetric system when the system is required to possess a prescribed set of first-order imaging properties [11]. The case we are considering here is first order only in the spatial coordinates but full order in *p*, *q*, and *p*´, *q*´. Unlike the rotationally symmetric case, now the coordinate systems at the object and image space have an arbitrary location and orientation one with respect the other, i.e., they do not have to share the *z*, *z*´ axis and nor they need to have parallel *x*, *x*´ and *y*, *y*´ axes.

Consider an example of optical system formed by 2 mirrors as shown in Fig. 1 (Left), and let’s try to design an aplanatic system with the same magnification factor *M* (*M = M*_{Y} = *M*_{X}) for both axes (a similar analysis can be done by substituting one or both mirrors by refractive surfaces) and with *p*_{0}´ = *q*_{0}´ = 0. We will investigate this configuration to see if it can constitute a freeform aplanatic system. Let’s call *L* as the optical path length of the rays connecting **O** and **O´**, which are the origins of the object and image planes (the orientation of the object coordinate systems with respect to the one at the image space is arbitrary).

The nomenclature in this treatment follows the notation shown in Fig. 1, bold letters refer to vectors (e.g. **N**), small bold letters as **u** refers to unit vectors, while italic refers to a scalar as *u* = (*x*^{2} + *y*^{2} + *z*^{2})^{½}. Consider an arbitrary point (*x*,*y*,*z*) on the first mirror. Using the path length condition, the distance of the image origin *u´* from the second optic can be deduced.

Now, let’s calculate the normal **N** at a point on the first mirror using the reflection law:

Note that **u** = (*x*,*y*,*z*)/*u*, and because of the Abbe sine condition *M p*´ = *p* and *M q*´ = *q*, i.e., *M* **u´**·**x´** = **u**·**x** and *M* **u´**·**y´** = **u**·**y**. Then **u´** can be written as **u´** = (*x* **x´** + *y* **y´**)/(*Mu*) + *g* **z´**, where the *g*(*x,y,z*) is making |**u´**| = 1, i.e., *g*^{2} = 1-(*x*^{2} + *y*^{2})/(*Mu*)^{2}. Since *u, u*´, **u**, **u´** can be written as functions of (*x*,*y*,*z*) and **OO´**, **x´**, **y´**, **z´** are constants, then **N** in [Eq. (2)] can be written as a function **N**(*x*,*y*,*z*), (**N** is then a vector field). If a solution of this 2-surface design problem exists, then this vector field must be integrable. The integrability condition is **N**·*∇* × **N** = 0 (*∇* × **N≡** curl **N**). We do not have degrees of freedom to impose the integrability condition on **N**. In Fig. 1 (Right) we have plotted the function|**N**·*∇* × **N**| for the above example. This function is equal to 0 only at a plane in this example. Thus, there is no first mirror solving the problem, i.e., in general, the problem has no solution with just two surfaces. Then, we conclude that *freeform aplanatism* cannot be achieved, in general, using two surfaces. Of course there are notable exceptions to this rule. In particular, in rotationally symmetric problems (around the *z* axis) the vectors **u**, **N** and **z** are contained in meridian planes and *∇* × **N** is perpendicular to **N** and then the integrability condition **N**·*∇* × **N** = 0 is fulfilled everywhere, which just confirms that the rotationally symmetric aplanats need only 2 surfaces, but can only solve problems with rotational symmetric constraints. This result is well known from previous works [8]. The two aplanatic points of any refractive sphere gives us a notable example of a single surface aplanatic system [12]. There are also notable exceptions among pure freeform devices: In [13] examples of 2 freeform surface afocal aplanatic systems are given.

It may seem surprising that freeform aplanats need in general 3 surfaces while rotational symmetric ones need just 2. This is because we are not comparing same things: A freeform aplanat comprises a set of conditions broader than that achievable by a rotational symmetric aplanat. For instance there is no solution for a rotational symmetric aplanat with different magnifications in *x* and *y* while the problem is in general solvable for the freeform case. We can also view the above scenario from the point of view of an SMS design. We have already established that, freeform aplanatic systems can be seen as a limiting case of SMS design method for three coincident points in [9]. The three (non collinear) points in the object space define the object plane and their images define the image plane. It is well known that for a *k*-point SMS design (where the footprint of the design ray bundles occupies the entire optical surface), we require *k* optical surfaces. Following the same logic, we also conclude that for a freeform aplanatic system, seen as a limit case of a three point SMS design, we require a minimum of three freeform optical surfaces, which at the end is caused by the need of at least three points to define the object (or image) plane.

From the design point of view, the rotational symmetric aplanat is a 2D geometry design (all the rays involved in the design are contained in a meridian plane). In 2D geometry only two points are needed to define the object or image straight lines, and consequently the SMS designs converging into aplanat designs need only two optical surfaces.

## 4. Semi aplanatic systems

It is possible to design a freeform system, with just 2 optical surfaces, which fulfills a single scalar equation relating the coordinates *p*, *q*, *p*´, *q*´: For instance we can choose one of the equations defining the Abbe sine condition for freeform prescription, i.e., *p*´ = *p*_{0}´ + *p/M*_{X} or *q*´ = *q*_{0}´ + *q/M*_{Y}, but not both. Observe that in this case one of the components of **u´** (either *p*´ or *q*´) is not determined as a function of (*x*,*y*,*z*) and we can use this degree of freedom to force **N**·*∇* × **N** = 0. We call such designs as semi-aplanatic systems. From the point of view of an SMS design, a semi-aplanatic design corresponds to the limiting case for two coincident points. These two points define one straight line in the object plane and a corresponding one in the image plane. Except for the points along that straight line, aberrations have linear dependency over the distance to the origin at the object plane, i.e., except for the points along that straight line the design is not aplanatic. An instance of this is described later in this paper.

The rotational symmetric case is not a semi-aplanatic case but a full aplanat one even though it needs just two surfaces. The 2 equations of the Abbe sine condition for freeform prescription become coincident for the rotational symmetric problems and hence the need of just 2 surfaces.

## 5. Three surface freeform aplanatic system formulation

This section will be devoted to the differential formulation of a three surface “freeform” aplanat. The concept of using a system of differential equations to determine an aplanatic rotational symmetric system was first used by Schwarzschild [8] and later by Wassermann and Wolf [14] and many others. We introduce an intermediate optic, as shown in Fig. 2, (surface containing point **M**) to the configuration in Fig. 1, while preserving the same nomenclature.

Using the same rules as before we will call *v***v** as the vector **AM** and *v*´**v´** as the vector **A´M**. We consider *u*,*v*,*u´,v´*,**u**,**v**,**u´**,**v´** functions of the initial direction cosines (*p*,*q*). For example, **u** = (*p*,*q,r*) where *r*^{2} *=* 1-*p*^{2}-*q*^{2} is the third direction cosine. All the vectors are addressed in the *x*,*y*,*z* coordinate system. All these functions but **u** are unknown functions of *p*, *q*. Since the aplanatic system must image **O** in **O´**, then the following two conditions are fulfilled (optical path length from **O** to **O**´ and relative position between **O** and **O´**):

Using Eq. (3) (4 scalar equations) we can reduce the list of unknown functions to *u*, *u´,* **v**, **u´** (for instance). The conditions for aplanatism are [9]:

Since **|u´|** = 1 and the vectors **x´** and **y´** are assumed to be known, these 2 equations determine **u´** as function of *p*, *q*, so the list of unknown functions may reduce to *u*, *u´,* **v** (remember that *M*_{X}, *M*_{Y}, *p*_{0}´ and *q*_{0}´are constants). Subsequently, as explained in [13] we can set the reflection law using Herzberger’s formulation as (first line is reflection at point **A**, and second at **A´**, sub-indices *p* and *q* denote partial derivatives.):

Reflection at **M** is implicit in the constant optical path length equation once the other 2 reflections are imposed. These set of 4 (scalar) partial differential equations along with the corresponding contour conditions, determine the last unknown functions (remember that **v** is a unit vector) with which the 3 surface aplanatic system can be calculated. The present formulation can be easily extended to refracting surfaces by just taking into account the refractive indices in the optical path length calculation [Eq. (3)], in the optical direction cosines *p*, *q*, *p*´, *q*´ and in both deflections of [Eq. (5)].

## 6. Freeform aplanatism – examples

The first example shown in Fig. 3 is a semi-aplanatic system composed of two freeform mirrors. It was designed such that the rays emanating from **O** perform stigmatic imaging from **O** to **O**´ and are forced to satisfy the equation *p*´ = *p*_{0}´ + *p/ M*_{X} (*M*_{X} = 0.5, *p*_{0}´ = 0) but not the equation *q*´ = *q*_{0}´ + *q/ M*_{Y}.

As we can see from Fig. 3, the distribution of RMS spot size at the image is parabolic for object positions along *x* direction exhibiting aplanatic characteristics, whereas it has a V shape cross-section along *y* direction as indicated by the facets in the plot. Additionally, we have evaluated the adherence of the equation of the Abbe sine condition used in the design and for the above system the quantity: |*p*´ - *p*_{0}´- *p*/ *M*_{X} |<1.435 10^{−8} for the rays emanating from **O**. We call these kind of systems as “semi aplanatic” alluding to the fact that it is aplanatic only in one direction.

The following system shown in Fig. 4 was designed for *M*_{X} = 0.2, *p*_{0}´ *=* 0, *M*_{Y} = 2.85 and *q*_{0}´ *=* −0.40674. The entrance pupil diameter was set at 350mm. The sizes of the three mirrors are as follows (1) Primary mirror: 275 mm (circular aperture) (2) Secondary mirror: 465.0798 mm (circular aperture) (3) Tertiary mirror: 503.0588 mm in *y* direction and 100 mm in *x* direction (for better representation). As indicated by the magnification factors, we can see the ray bundle expanded along the *y* direction and compressed in the *x* direction. It is a 3 freeform mirror aplanatic design. We can also see from the distribution of RMS spots along two directions of the object position that the system has a parabolic behavior along any direction. Additionally, we have evaluated the adherence of Abbe sine condition and for this system the quantity: |*p*´- *p*_{0}´- *p*/ *M*_{X} |<4 10^{−10}; |*q*´- *q*_{0}´- *q*/ *M*_{Y} |< 3 10^{−9}. We call these kinds of systems as aplanatic or “full aplanatic” alluding to the fact that it is aplanatic in any direction.

## 7. Conclusions

A freeform aplanatic design is an optical system stigmatically imaging the origin of the object plane to the origin of the image plane and fulfilling the freeform Abbe sine condition. This condition is constituted by two scalar equations (unlike the usual rotational symmetric Abbe sine condition which has only one). We have proved how two optical surfaces are in general insufficient for achieving freeform aplanatism, through the use of the integrability condition. We have presented some illustrative examples using Code V^{®} which are in agreement with basic laws governing aplanatism. A two surface design can in general fulfill just one of the 2 equations defining the freeform Abbe sine condition. We have called such 2 freeform surface designs as semi-aplanatic.

## Funding

European Commission (ADOPSYS: FP7-PEOPLE-2013-ITN 608082), Spanish Ministries (OPTIVAR: TEC2014-56867-R, GUAKS: RTC-2014-2091-7, SHIVA: RTC-2016-5295-7), UPM (Q090935C59).

## References and links

**1. **J. Rolland and K. Thompson, “Freeform optics: Evolution? no, revolution!” SPIE Newsroom (2012).

**2. **F. Duerr, Y. Meuret, and H. Thienpont, “Potential benefits of free-form optics in on-axis imaging applications with high aspect ratio,” Opt. Express **21**(25), 31072–31081 (2013). [CrossRef] [PubMed]

**3. **H. Ries, N. E. Shatz, J. C. Bortz, and W. Spirkl, “Consequences of skewness conservation for rotationally symmetric nonimaging devices,” Proc. SPIE **3139**, 47–58 (1997). [CrossRef]

**4. **J. C. Miñano, P. Benítez, and A. Santamaría, “Free-form optics for illumination,” Opt. Rev. **16**(2), 99–102 (2009). [CrossRef]

**5. **E. Abbe, “Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung,” Archiv für mikroskopische Anatomie 9, pp. 413–468, (1873).

**6. **T. T. Elazhary, P. Zhou, C. Zhao, and J. H. Burge, “Generalized sine condition,” Appl. Opt. **54**(16), 5037–5049 (2015). [CrossRef] [PubMed]

**7. **J. J. Braat and P. F. Greve, “Aplanatic optical system containing two aspheric surfaces,” Appl. Opt. **18**(13), 2187–2191 (1979). [CrossRef] [PubMed]

**8. **K. Schwarzschild, “Untersuchungen zur geometrischen Optik II,” Abh. Konigl. Ges. Wis. Gottingen Mathphys. Kl. **4**, 1–3 (1905).

**9. **J. C. Miñano, P. Benítez, and B. Narasimhan, “Freeform aplanatic systems as a limiting case of SMS,” Opt. Express **24**(12), 13173–13178 (2016). [CrossRef] [PubMed]

**10. **J. H. Burge, C. Zhao, and S. H. Lu, “Use of the Abbe sine condition to quantify alignment aberrations in optical imaging systems,” Proc. SPIE **7652**, 765219 (2010). [CrossRef]

**11. **B. D. Stone and G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A **9**(3), 478 (1992). [CrossRef]

**12. **R. K. Luneburg, *Mathematical Theory of Optics* (University of California, Los Angeles, 1964).

**13. **P. Benitez, M. Nikolic, and J. C. Miñano, “Analytical solution of an afocal two freeform mirror design problem,” Opt. Express **25**(4), 4155–4161 (2017). [CrossRef] [PubMed]

**14. **G. Wassermann and E. Wolf, “On the theory of aplanatic aspheric systems,” Proc. Phys. Soc. B **62**(1), 2–8 (1949). [CrossRef]