## Abstract

Freeform surfaces play important roles in improving the imaging performance of off-axis optical systems. However, for some systems with high requirements in specifications, the structure of the freeform surfaces could be very complicated and the number of freeform surfaces could be large. That brings challenges in fabrication and increases the cost. Therefore, to achieve a good initial system with minimum aberrations and reasonable structure before implementing freeform surfaces is essential for optical designers. The already existing initial system design methods are limited to certain types of systems. A universal tool or method to achieve a good initial system efficiently is very important. In this paper, based on the Nodal aberration theory and the system design method using Gaussian Brackets, the initial system design method is extended from rotationally symmetric systems to general non-rotationally symmetric systems. The design steps are introduced and on this basis, two off-axis three-mirror systems are pre-designed using spherical shape surfaces. The primary aberrations are minimized using the nonlinear least-squares solver. This work provides insight and guidance for initial system design of off-axis mirror systems.

© 2017 Optical Society of America

## 1. Introduction

In reality due to several reasons, the major number of optical systems are rotationally symmetric. However, in recent years, systems without rotational symmetry are investigated in specialized applications. For instance, unobscured three mirror anastigmats (TMA) are designed to achieve high resolution and small system size [1–6]. Scheimpflug systems maintain rotationally symmetric lens elements but with asymmetric imaging condition [7–9]. Both TMAs and Scheimpflug systems have one-dimension symmetry. Some applications combine reflective and refractive elements such as head-mounted display (HMD) systems [10–12]. In order to design a system, in addition to design methods, the corresponding aberration theory is necessary to analyze the system performance. For rotationally symmetric systems, Seidel aberration theory is often used as a tool in optimization and analysis of the aberration contributions of each surface in the system. The design strategy can be guided following the aberration contributions. However, for off-axis systems or general non-symmetrical systems, the traditional Seidel aberration theory is no longer valid due to the off-axis contribution from each surface. Additionally, aspheres and freeform surfaces provide great possibilities to obtain a high performance in resolution in the non-rotationally symmetric systems [3, 10, 11, 13–18]. This leads to other questions, such as how many freeform surfaces are necessary for one system, which the best position of the freeform surface is.

The initial system typically influences the number and complexity of freeform surfaces, which correspond to the cost and difficulty in fabrication. Thus, an initial system with minimum aberrations is preferred. The conventional method to design an initial system often begins with the thin-lens model [19]. It works fine with even complicated systems such as zoom systems. Nevertheless, non-rotationally symmetric systems are more complicated. The orientation and curvature of the surfaces impact tremendously on the aberrations. The thin lens model can guarantee astigmatism correction and the first order properties such as focal power in a near neighborhood of the axis ray. However, when substituting the thin lenses with real lenses, it is hard to obtain a solution for non-rotationally symmetric systems without obscuration. There are also new methods to obtain specific initial systems. Cartesian surfaces are used to obtain mirror systems. If the geometric focal points coincide with each other, the central field can be perfectly imaged, which means the nodal point is shifted to the center of the field [3, 4, 20]. However, the problems appear when introducing the freeform surface due to the large off-axis use of the surfaces. In case of finite field of view, the SMS method is a direct design method using freeform surfaces. This approach allows coupling of rays from a certain number of fields into image points by using freeform surfaces [13, 14]. The limitation currently is also the number of freeform surfaces that the design procedure deal with. All the methods above provide limited ability in the analysis of the system during the design procedure due to extended field of view and broadband illumination. It is well known that the aberration contributions of each element influence the sensitivity of the whole system. Based on the Gaussian brackets investigated by Tanaka, the Generalized Gaussian Constants (GGC’s) give the paraxial ray tracing data fast in matrix computation, which provides a possibility to optimize the Seidel aberration coefficients based on paraxial ray tracing data. The first order properties can be also analytically fulfilled using GGC’s [21]. In this way, the initial configuration is designed analytically using the Seidel aberration theory [22]. However, traditional Seidel aberration coefficients can only describe the primary aberrations of rotationally symmetric systems. Therefore, to extend this design method to general systems, the traditional Seidel aberration theory should be substituted by more general aberration theory.

In this paper, we extended the Gaussian brackets design method to non-rotationally symmetric systems. This method is based on Nodal aberration theory invented by Kevin Thompson, which includes the influence of the tilt and decenter of each surface of the system [5, 6, 23, 24]. Nodal aberration theory is firstly invented concerning systems with rotationally symmetric surfaces. In recent years, it is extended to freeform surfaces [15–17]. In this method, aberrations of the central field are optimized analytically in the first approach using spherical surfaces, which try to bring the nodal points to the center of the field of view. The first order properties and the stop location can be also fulfilled. The aberration theory and the method are discussed and explained in detail in Sec. 2. Two kinds of three mirror anastigmats are designed to demonstrate the proposed design method in Sec. 3. Finally, in Sec. 4 the advantages and shortcomings of this method are discussed and the outlook is provided.

## 2. Initial system design with spherical surfaces based on Gaussian brackets and Nodal aberration theory

#### 2.1 Definition of Generalized Gaussian Constants

According to the description of Gaussian brackets [21] formulated by TANAKA, a Gaussian bracket, whose elements consist of a set of numbers or functions, ${a}_{i},{a}_{i+1},{a}_{i+2},\cdots ,{a}_{j-1},{a}_{j}$, is written in the form as Eq. (1).

In an optical system with the focal power ${\Phi}_{i}$of the i^{th} component, the reduced distance $e{\text{'}}_{i}$between the i^{th} and the (i + 1)^{th} components, based on Gaussian brackets definition, Generalized Gaussian Constants (GGC’s) [21, 22] are defined to formulate the paraxial theory of optical systems. The Generalized Gaussian Constants for the subsystem from i^{th} to j^{th} components are expressed with Gaussian brackets as Eqs. (2)-(5).

For systems consisting of only spherical surfaces, the focal power and the reduced distance are

Where ${c}_{i}$ is the curvature of the i^{th} surface; ${n}_{i-1}$ is the refractive index before the i^{th} surface; ${n}_{i}$ is the refractive index after the i^{th} surface; ${d}_{i}$ is the thickness from the i^{th} surface to the (i + 1)^{th} surface.

The relation between those four constants are as Eqs. (8)-(11).

Therefore, the paraxial ray tracing can be represented using the following four formulas.

The first order properties of the system with k surfaces are

#### 2.2 Nodal aberration theory of non-rotationally symmetric systems with spherical surfaces

According to Nodal aberration theory introduced by Thompson [5, 6, 15–17, 23, 24], the wave aberrations for non-rotationally symmetric systems are built upon a vectorial formulation. The field decenter contribution of each surface should be described by introducing a displacement vector ${\overrightarrow{\sigma}}_{j}$.

The wave aberration expansion is represented by vectors:

Where $\overrightarrow{H}$ denotes the normalized vector for the field height in the image plane. ${\overrightarrow{\sigma}}_{j}$ denotes the field decenter vector of each surface. $\overrightarrow{\rho}$ denotes a normalized vector describing the pupil height.

The wave aberration expansion to the fourth order including tilts and decenters can be expressed as

${W}_{klm}$ can be obtained using the on-axis centered system paraxial raytrace data. The marginal ray angle ${u}_{j}$, the chief ray angle ${\overline{u}}_{j}$, the marginal ray height ${h}_{j}$, and the chief ray height ${\overline{h}}_{j}$ are calculated analytically using Generalized Gaussian Constants. Because the field decenter vector has no impact on the five Seidel coefficients presented in Eqs. (20)-(24), the paraxial ray tracing procedure is regarded as in the on-axis system.

Therefore, we obtain the aberration coefficients of primary monochromatic aberrations in non-rotationally symmetric systems with spherical surfaces as in Table 1. Since we firstly deal with the simple case of mirror systems, the two primary chromatic aberration coefficients are not considered in this paper.

If the system consists of only spherical surfaces, the decenter and tilt of one surface can always be defined as an equivalent tilt parameter because the vertex can be defined at an arbitrary position of the surface. The corresponding real ray based field decenter vector ${\overrightarrow{\sigma}}_{j}$ is given as Eq. (25):

where ${\overrightarrow{N}}_{j}$ denotes the unit normal vector of the object plane and of its conjugates; ${\overrightarrow{R}}_{j}$ denotes the unit direction vector of the optical axis ray (OAR); ${\overrightarrow{S}}_{j}$ denotes the unit normal vector of the surface at the OAR intersection point [24].

Here, we deal with the simple case that the object plane and its conjugates are always perpendicular to optical axis ray, which means certain types of systems such as Scheimpflug systems are excluded. The three direction vectors are defined in the local coordinate of each incident optical axis ray as

The vector ${\overrightarrow{\sigma}}_{j}$ can then be derived as

#### 2.3 Conversion of surface normal vector into Euler angles

In this paper, the optical axis ray is defined as the chief ray of the central field ($\overrightarrow{H}=0$). If we regard the intersection point of the optical axis ray at each surface as the vertex of the surface, the tilt of the surface can be calculated using the tilt of the optical axis ray. The three components of the surface normal vector ${\overrightarrow{S}}_{j}$ should be convertible with Euler angles to obtain the coordinate tilts in the system.

Here we mainly consider plane-symmetric systems. For plane-symmetric reflective systems, the tilt angle of the j^{th} surface around the vertex point is equal to the optical axis ray incident angle ${i}_{j}$, which is also equal to the angle between ${\overrightarrow{R}}_{j}$ and ${\overrightarrow{S}}_{j}$, as shown in Fig. 1. After the surface, the coordinate should be tilted by reflection angle $i{\text{'}}_{j}$ to keep the optical axis along the optical axis ray.

The tilt angles of the optical axis ray can be derived as

In plane-symmetric refractive system as shown in Fig. 2, the first coordinate tilt before the j^{th} surface is the same as in reflective system, which is represented by the incident angle of the optical axis ray. After refraction, in order to keep the optical axis along the optical axis ray, the second coordinate break after the surface should be tilted by an angle $i{\text{'}}_{j}$, which is calculated by refraction law.

The vectors ${\overrightarrow{R}}_{j}$ and ${\overrightarrow{S}}_{j}$ are presented the same as in Eqs. (30)-(31). The tilt angle ${i}_{j}$ of the first coordinate break before one surface is the same as in Eq. (32). The second coordinate break after the surface should be tilted by angle $i{\text{'}}_{j}$, which is derived as

#### 2.4 Coddington equations and Petzval curvature

In non-rotationally symmetric systems, due to the incident angles of the chief rays at each surface, the focal power is different in tangential and sagittal planes. The astigmatism term in Nodal aberration theory contains focal shift, transverse displacement from the optical axis ray, and the astigmatism with respect to the medial surface. Therefore, to control the astigmatism of the central field in a more direct way, we apply Coddington equations as an additional constraint in the method. However, it is valid for spherical surfaces. In the future, when applying freeform surfaces to the method, the astigmatism in aberration theory with x and y components is important. If the chief ray incident angle at a surface is presented by $\overline{i}$, the Coddington equations are as

The sagittal imaging

The tangential imaging

The local focal power ${\Phi}_{\text{skew}}$ of each surface is defined as

For each surface, we can derive the object distance and image distance respectively in sagittal and tangential plane as

Equation (39) is regarded as the condition to minimize the difference between the sagittal and tangential image distances after the last surface k.

In this method, the aberrations of the central field ($\overrightarrow{H}=0$) are derived by the GGC’s and the field decenter vectors ${\overrightarrow{\sigma}}_{j}$, but field curvature characterize the whole field. Therefore, we add one more condition based on the Petzval sum of the system as Eq. (40) to vanish the field curvature, which also reduces one degree of freedom. Here ${R}_{ptz}$ denotes the Petzval radius.

#### 2.5 Design procedure

In this paper, the initial design procedure is as in the following steps

- 1. Define the number of surfaces including the pupils
- 2. Define initial ray data
- 3. Define stop position
- 4. Apply the equation of Petzval curvature vanishing to present one surface curvature by the other curvatures
- 5. Define Gaussian brackets and derive the Generalized Gaussian constants
- 6. Fast on-axis paraxial ray trace by using GGC’s
- 7. Derive the Seidel aberration coefficients using the on-axis paraxial ray tracing data
- 8. Derive focal length and other first order properties by using GGC’s
- 9. Derive the 4th order Nodal aberration coefficients by adding the real ray based field decenter vectors as variables
- 10. Derive the sagittal and tangential image distances
- 11. Obtain the analytical functions
- 12. Minimize the functions by nonlinear least-squares solver in Matlab
- 13. Convert the solutions into system data and check the performance of the system in design software

In Section 2.1-2.4 the basis of step 4 to step 10 are introduced. In step 11, we obtain a series of nonlinear functions with the same unknown parameters analytically. In Table 2, the nonlinear functions are listed for a mirror system. Because the mirror systems have no chromatic aberration, only the five monochromatic primary aberrations are derived together with the first-order properties and the Coddington equations.

The optimization toolbox in Matlab provides the possibility of nonlinear fitting optimization. A solution which leads to the minimum value of a series of the nonlinear functions would be obtained. The upper limit and lower limit of the unknown parameters should be defined according to the pre-designed geometry of the system. If the optimization function cannot provide a good solution, the upper and lower limits together with the starting value should be readjusted until a good starting system is solved. This is the limitation of this method because the optimization tool provides only local minimum search possibility. The parameters in the solution of the nonlinear functions should be converted into parameters which can be imported into design software.

## 3. Initial system design of TMA systems

In this paper, we propose two TMA systems as examples to demonstrate the proposed new initial system design method. The reason to choose mirror systems is because they have no chromatic aberration, hence choice and optimization concerning glass are not necessary. Since the size of a TMA system usually has constraints and one of the main tasks is to avoid obscuration, we add one first order property to control the working distance in the nonlinear functions, which is the back focal length$S{\text{'}}_{F}$. Since we take the position of stop into consideration in this method, the position of stop and its intermediate images are also treated as surfaces in the system with zero focal power and no tilts. Therefore, the on-axis model of a TMA system can be shown as in Fig. 3.

The TMA system in Fig. 3 consists of eight surfaces in total including three mirrors, pupil with its intermediate images, and the image plane. The focal power of the intermediate pupils and the image plane is zero.

${t}_{j}$ denotes the thickness from real surface to surface. Only ${t}_{1}$ denotes the thickness from the entrance pupil to the first mirror. They can be represented by the reduced distances $e{\text{'}}_{j}$. The focal power ${\Phi}_{j}$ of mirrors would be converted to radii of curvature of mirrors.

After the on-axis model is defined, the paraxial ray trace is performed to obtain the Seidel coefficients ${W}_{klm}$ using the parameters in Fig. 3. By adding the field decenter vector ${\overrightarrow{\sigma}}_{j}$, the functions as in Table 2 are derived analytically. The nonlinear functions are transferred to the nonlinear least-squares solver to obtain the solutions which leads to the minimum aberrations.

#### 3.1 Zigzag structure TMA system

The first example is the zigzag structure TMA system, for which we try to design with negative-positive-positive (NPP) structure by setting different boundary values in the optimization of the nonlinear functions. The design specifications are given in Table 3. The stop position is on the second mirror, which means the intermediate pupil 1 and intermediate pupil 2 in Fig. 3 coincide at the position of M2. The reduced distance $e{\text{'}}_{3}$ and $e{\text{'}}_{4}$ are zero.

The initial ray data are defined as in Table 4.

Using the Petzval curvature vanishing relation, the curvature of the last mirror can be expressed by the first two mirrors. The thicknesses ${t}_{2}$ can be expressed by the imaging relation of the entrance pupil by the first mirror. The unknown parameters are the three tilts of the mirrors, curvature of the first two mirrors, and two thicknesses ${t}_{1}$ and ${t}_{3}$.

Due to the plane-symmetry of the TMA system, there is only the y-direction component of the field decenter vector. Therefore, some of the aberrations of the central field are zero, for instance, the coma in x direction. Hence, there are totally nine nonlinear functions with seven unknown parameters. The nonlinear least-squares solver performs a local minimum search. It can be used with optional lower and upper bounds. Similar with other local optimization algorithms, the starting value is critical to obtain a good starting system. It is possible that a minimum value is not found when the iteration number exceeds the limit setting in the toolbox. After several readjustments of the boundary values and the starting values, we worked out the design results with no obscuration.

The boundary values and the solutions after nonlinear function optimization are given in Table 5 to obtain the initial system. The sign and boundary value of the tilts are defined in the range to avoid obscuration. When the system is designed to be zigzag structure, the first and third tilts are positive and the second tilt is negative.

The layout, the spot diagram and the RMS spot radius over the whole FOV of the zigzag structured initial system are illustrated in Fig. 4. Corresponding astigmatism, coma, and distortion over the FOV are shown in Fig. 5. The performance is evaluated by the spot size which is uniform over the whole field. One nodal point of astigmatism can be seen in Fig. 5a. The other nodal point is outside of the field of view due to the boundary conditions to achieve no obscuration. Coma is dominated by the field invariant part. Distortion is −1.5%.

#### 3.2 Compact folding structure TMA system

The second example is designed with small f-number. The specifications are obtained from the proceeding of H. Zhu [3]. Here we use spherical surfaces to establish the initial system and the vertex of the surface is at the intersection point of the central field chief ray, which would be an advantage for further optimization with freeform surfaces. The entrance pupil diameter and the focal length are both very large. Therefore, a folding structure is normally used to reduce the size of the system. The specifications are given in Table 6.

The initial ray data are defined as in Table 7.

In this case ${t}_{2}$ cannot be expressed by any other parameters. Therefore, ${t}_{2}$ is also one of the unknown parameters. The boundary values and the solutions after nonlinear function optimization are presented in Table 8. To obtain compact folding structure, the three tilts are all positive.

The layout, the spot diagram and the RMS spot radius over the whole FOV of compact folding structure initial system are illustrated in Fig. 6. The spot size is also uniform. Corresponding astigmatism, coma, and distortion over the FOV are shown in Fig. 7. The second nodal point for astigmatism is also out of the field of view. However, in this system, the field depend coma has larger influence on the total coma. The system suffers from keystone distortion.

## 4. Conclusion

In this paper, an initial system design method based on Gaussian brackets and Nodal aberration theory is proposed for non-rotationally symmetric systems. The primary aberrations and first order properties can be presented analytically and optimized by nonlinear function optimization tool of Matlab. The stop position is taken into consideration in this method. Obscuration can be avoided by setting proper boundary value of the unknown parameters. However, this method is limited to a global minimum searching ability due to the limitation of the nonlinear function optimization tool. In the future, the method can be applied to more complicated systems with tilted object such as Scheimpflug systems. The optimization can be extended from a single central field to more fields and the aberration contribution of each surface can be also taken into consideration. Nodal aberration theory is recently extended to describe the aberrations of freeform surfaces. Therefore, this method can also be extended to initial system design with freeform surfaces by the fourth-order Zernike polynomial terms. In the future work, it is planned to make systematic comparisons between different initial system design methods. Since the methods are suitable for different systems and have own limitations, a careful choice of system type, performance criteria and the strategy of comparison are important aspects to be taken into consideration.

## Funding

Bundesministerium für Bildung und Forschung (BMBF) (03WKCK1D); Bundesministerium für Bildung und Forschung (BMBF) (031PT609X).

## Acknowledgments

We thank our colleagues who provide insight and expertise that greatly assisted the research.

## References and links

**1. **H. J. Juranek, R. Sand, J. Schweizer, B. Harnisch, B. P. Kunkel, E. Schmidt, A. Litzelmann, F. Schillke, and G. Dempewolf, “Off-axis telescopes: the future generation of Earth observation telescopes,” Proc. SPIE **3439**, 104–115 (1998). [CrossRef]

**2. **D. E. Lencioni, C. J. Digenis, W. E. Bicknell, D. R. Hearn, and J. A. Mendenhall, “Design and performance of the EO-1 Advanced Land Imager,” Proc. SPIE **3870**, 269–280 (1999). [CrossRef]

**3. **H. Zhu, Q. Cui, M. Piao, and C. Zhao, “Design of a dual-band MWIR/LWIR circular unobscured three-mirror optical system with Zernike polynomial surfaces,” Proc. SPIE **9272**, 92720W (2014). [CrossRef]

**4. **S. Chang, “Linear astigmatism of confocal off-axis reflective imaging systems with N-conic mirrors and its elimination,” J. Opt. Soc. Am. A **32**(5), 852–859 (2015). [CrossRef] [PubMed]

**5. **K. P. Thompson, T. Schmid, and J. P. Rolland, “The misalignment induced aberrations of TMA telescopes,” Opt. Express **16**(25), 20345–20353 (2008). [CrossRef] [PubMed]

**6. **K. P. Thompson, K. Fuerschbach, T. Schmid, and J. P. Rolland, “Using nodal aberration theory to understand the aberrations of multiple unobscured three mirror anastigmatic (TMA) telescopes,” Proc. SPIE **7433**, 74330B (2009). [CrossRef]

**7. **H. Gross, *Handbook of Optical Systems* (Wiley-VCH, 2005), Vol. 1.

**8. **H. Mayer, “Theodor Scheimpflug,” Ophthalmic Res. **26**(1), 3–9 (1994). [CrossRef]

**9. **C. Bakolias and A. K. Forrest, “Dark-field Scheimpflug imaging for surface inspection,” Proc. SPIE **3029**, 57–68 (1997). [CrossRef]

**10. **D. Cheng, Y. Wang, H. Hua, and J. Sasian, “Design of a wide-angle, lightweight head-mounted display using free-form optics tiling,” Opt. Lett. **36**(11), 2098–2100 (2011). [CrossRef] [PubMed]

**11. **D. Cheng, Y. Wang, C. Xu, W. Song, and G. Jin, “Design of an ultra-thin near-eye display with geometrical waveguide and freeform optics,” Opt. Express **22**(17), 20705–20719 (2014). [CrossRef] [PubMed]

**12. **H. Hua, X. Hu, and C. Gao, “A high-resolution optical see-through head-mounted display with eyetracking capability,” Opt. Express **21**(25), 30993–30998 (2013). [CrossRef] [PubMed]

**13. **R. A. Hicks, “Direct methods for freeform surface design,” Proc. SPIE **6668**, 666802 (2007). [CrossRef]

**14. **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**(26), 24036–24044 (2009). [CrossRef] [PubMed]

**15. **K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Extending Nodal Aberration Theory to include mount-induced aberrations with application to freeform surfaces,” Opt. Express **20**(18), 20139–20155 (2012). [CrossRef] [PubMed]

**16. **T. Yang, J. Zhu, and G. Jin, “Nodal aberration properties of coaxial imaging systems using Zernike polynomial surfaces,” J. Opt. Soc. Am. A **32**(5), 822–836 (2015). [CrossRef] [PubMed]

**17. **H. Shi, H. Jiang, X. Zhang, C. Wang, and T. Liu, “Analysis of nodal aberration properties in off-axis freeform system design,” Appl. Opt. **55**(24), 6782–6790 (2016). [CrossRef] [PubMed]

**18. **O. Cakmakci, B. Moore, H. Foroosh, and J. P. Rolland, “Optimal local shape description for rotationally non-symmetric optical surface design and analysis,” Opt. Express **16**(3), 1583–1589 (2008). [CrossRef] [PubMed]

**19. **R. Kingslake and R. B. Johnson, *Lens Design Fundamentals* (Academic Press, 2009).

**20. **D. Korsch, *Reflective Optics* (Academic Press, 1991).

**21. **K. Tanaka, “II Paraxial theory in optical design in terms of Gaussian Brackets,” Prog. Opt. **23**, 63–111 (1986). [CrossRef]

**22. **X. Yuan and X. Cheng, “Lens design based on lens form parameters using Gaussian brackets,” Proc. SPIE **9272**, 92721L (2014). [CrossRef]

**23. **K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” The University of Arizona (1980).

**24. **K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A **26**(6), 1503–1517 (2009). [CrossRef] [PubMed]