Generalized Aldis theorem for calculating aberration contributions in freeform systems

Bo Chen; Alois M. Herkommer

doi:10.1364/OE.24.026999

1. Introduction

In optics industry still the most frequently used lens shapes are rotational symmetric spherical and aspherical lenses, since they can easily be manufactured, assembled and tested. However, with the development of high precision diamond turning machines and the employment of plastics or printed optics, freeform surface shapes are increasingly applied in a variety of situations [1–3]. For instance, freeform surfaces are used in head mounted displays to make them compact and reduce the weight [2–4]. Application of freeform surfaces in three mirror anastigmats [5], view finders [6], and catadioptric systems [7] can enlarge the field of view and minimize the aberrations in these systems. Additionally, freeforms can also be applied to boost rotational symmetric systems to good image quality even for high aspect ratio [8].

As freeform system per definition do not allow any assumptions on symmetry, the well-developed aberration theories of rotational cases, for instance Aldis calculus [9–11], or Seidel aberration theory [12,13] cannot be applied. In contrast the general vector aberration method or nodal aberration theory [14,15] allow a mathematical treatment of non-symmetric aberration fields. However these approaches are quite complex, require in-depth analysis of the system and it is difficult to extract individual surface aberration contributions. Therefore it is desirable to find alternate methods to easily calculate contributions from all surfaces, just like Seidel or Aldis aberrations do in rotational symmetric cases. By acquiring those aberration contributions, the designer is allowed to focus on strongly contributing surfaces, balance individual aberrations, or estimate the sensitivity of a freeform design in a similar way as for standard systems.

The method of phase space in optics [16,17] allows an alternate treatment of optical systems, employing an illustration of the system behavior via the position and angle of rays. This has proven to be especially helpful for paraxial imaging systems [18] and for illumination problems [19,20]. Aberrations in this picture correspond to non-linear transformations within phase space, which allows an alternate access to aberrations within freeform system [21,22]. A quantitative investigation for rotational symmetric systems of the authors recently has shown, that analysis of a pair of differential rays in phase space compares well to the exact calculation of Aldis theory [23]. Since symmetry assumptions were not required for this calculation, we will in this work investigate the potential of this method for freeform systems.

2. Phase space in non-rotational symmetric systems

2.1. Concept of phase space and definitions

In general the concept of phase space in geometrical optics is based on an illustration of ray angles versus ray positions. In the paraxial regime this is closely related to the so-called matrix-optics or ABCD formalism [16–18]. In illumination design non-paraxial definitions, employing the exact direction cosines, allow to illustrate etendue and radiance transport [19,20]. It was already shown that a comparison of the paraxial and the exact ray-tracing behavior allows to calculate surface aberration contributions in rotational symmetric systems [21]. This paper will follow up along these lines and mathematical prove the validity of this method for all optical systems, where a reference ray can be defined, in particular freeform systems.

Since for freeform systems rotational symmetry cannot be assumed, the position and direction of each geometrical ray in a certain reference plane has to be described by four quantities, namely r = (x,y,u,v)^T. Here x,y are the intersection points of the ray with the reference surface and u,v are the index-weighted direction tangents, which can easily be computed from the normalized direction vector (L, M, N) and the refractive index n by using

v = n \cdot \frac{M}{N} u = n \cdot \frac{L}{N} .

It is important to note, that the exact ray direction tangents and not any “paraxial” angles are used here (Fig. 1). As a consequence of this definition any free ray-propagation between parallel reference planes D_{i + 1} and D’_i, separated in the z-direction by some distance d, is exactly identical to the linear propagation matrix, which is known from paraxial ABCD-calculus.

r_{i + 1} = T_{i} \cdot r'_{i} where T_{i} = (\begin{matrix} I & d \cdot I \\ 0 & I \end{matrix}) .

Here I is the two-dimensional unit matrix.

Fig. 1 Ray definition and propagation in between parallel reference planes.

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2.2. Aberration calculation

Same as in Aldis theory [9,10] the aberration calculation is based on real ray tracing and paraxial (linear) ray-tracing of a given input ray. Throughout this paper the real (exact) rays are denoted by r, and paraxial rays (after a real surface) are denoted by p. In order to trace, record and calculate any propagating ray a series of planar dummy surfaces D_i and D’_i before and sequentially after each real surface needs to be defined. Each dummy-surface is orthogonal to the optical axis, respectively to the reference ray in a tilted and decentered freeform system, as illustrated in Fig. 2. Each intersection of the reference ray with a real surface thus defines a pair of dummy surfaces both being centered and orthogonal on the reference ray intersection. By performing raytracing the real raytracing data r_i on each “input”-dummy surface D_i, and r’_i on the corresponding “output”-dummy surfaces D’_i can be recorded with the help of any ray-tracer.

Fig. 2 Illustration of ray propagation: a) defines the reference ray and the dummy surfaces, b) illustrates the ray propagation in the unfolded reference system of the dummy surfaces, c) illustrates the ray positions in phase space.

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In order to derive aberration contributions a comparison of the exact raytracing to the paraxial (linear) system behavior is required. According to Eq. (2) the propagation in between consecutive parallel dummy surfaces D’_i and D_{i + 1} is linear and therefore free of aberrations. Thus, as expected, the aberrations are induced by the action of the real surface in between each dummy surface pair D_i and D’_i. To derive the aberration contribution of the surface i, the real (nonlinear) transformation of the input ray r_i to the output ray r’_i is compared to the linear transformation into a paraxial output ray p’_i, which generally is given by:

p'_{i} = S_{i} \cdot r_{i} .

The linear matrix S_i can for spherical surfaces be calculated from the curvature and the refractive index, according to the well-known rules of ABCD-matrix methods. For the general case we can determine the matrix either from a set of rays close to the reference ray, and then solving a set of equations to find the best-fit linear four-dimensional matrix. Another option is to employ the ABCD-function in Synopsis CodeV [24], which is valid for decentered and tilted surfaces.

For every surface i thus the aberration vector between the real ray r and the paraxial ray p can be defined as δ_i = (r’_i-p’_i) and illustrated in Fig. 2. If we propagate this aberration vector forward to the image plane, we will get the corresponding aberration contribution at the image plane, denoted by Δ_i. In reference [23] we have already shown that for a rotational symmetric system those aberration contributions at the image are almost identical to the exact Aldis surface contribution, if we paraxially propagate the local aberration vector δ_i to the image, as

Δ_{i} = M_{I, i} \cdot (r'_{i} - p'_{i}) .

Here M_I,i represents the paraxial propagation matrix from the exiting dummy surface D’_i to the image plane. This matrix is given by:

M_{I, i} = T_{n} S_{n} \dots S_{i + 1} T_{i},

where the index n represents the last surface in the system, and T_n is the propagation from this last surface n to the image plane. For later use it is also important to note that this matrix follows a recursion formula as every matrix is the combination of the prior surfaces:

M_{I, i} = M_{I, i + 1} S_{i + 1} T_{i} .

We will now mathematical prove that the sum of all surface contributions is identical to the exact real ray aberrations of the system at the image. To do so we consider the aberrations of all surfaces, starting from surface i = 1 to n. The individual aberration contributions can be calculated as:

\begin{array}{l} Δ_{1} = M_{I, 1} \cdot (r'_{1} - p'_{1}), \\ Δ_{2} = M_{I, 2} \cdot (r'_{2} - p'_{2}), \\ ⋮ \\ Δ_{n} = M_{I, n} \cdot (r'_{n} - p'_{n}) . \end{array}

Let us consider the first term of each surface contribution. Due to the recursion relation, Eq. (7), above we can also express this term as:

M_{I, i} \cdot r'_{i} = M_{I, i + 1} S_{i + 1} \underset{r_{i + 1}}{\underset{︸}{T_{i} \cdot r'_{i}}} = M_{I, i + 1} \cdot p'_{i + 1},

where in the second step we have used the fact, that exact propagation via T_i of the real ray r’_i after surface i will result in the real input ray before the next surface r_i₊₁. Further paraxial propagation of this ray via surface S_i₊₁ results in the paraxial ray p’_i₊₁. We can apply this to modify the first term in all equations, besides the last, since the recursion for the last surface n is not applicable, as it is the last.

\begin{matrix} Δ_{1} = M_{I, 1} \cdot (r'_{1} - p'_{1}) = M_{I, 2} \cdot p'_{2} - M_{I, 1} \cdot p'_{1}, \\ Δ_{2} = M_{I, 2} \cdot (r'_{2} - p'_{2}) = M_{I, 3} \cdot p'_{2} - M_{I, 2} \cdot p'_{2}, \\ \dots \\ Δ_{n} = M_{I, n} \cdot (r'_{n} - p'_{n}) = M_{I, n} r'_{n} - M_{I, n} \cdot p'_{n} . \end{matrix}

Considering the sum of the aberrations of all surfaces in the system, we can see from the above equations that in every line the first term cancels with the second term of the next line. In total this only leaves the second term of the first line and the first term of the last line.

\sum_{i} Δ_{i} = M_{I, n} r'_{n} - M_{I, 1} p'_{1} .

To further simplify and understand the implications of this result we can make use of the fact that the first term of the result simply represents the real ray at the image plane, since M_I,n = T_n represents free propagation from the last dummy surface to the image, which is exact and thus we have M_I_,_nr’_n = T_nr’_n = r_I.

Also the second term can be simplified, since the paraxial ray coordinates after surface 1 directly result from the paraxial propagation of the input ray as M_I,₁p’₁ = M_I_,1 S₁T₀r₀ = M_I_,0r₀ = p_I. In summary therefore the sum of all aberrations can be simplified to:

\sum_{i} Δ_{i} = r_{I} - p_{I} .

The right hand side of this equation corresponds to the difference between an exact traced ray of the full system and a paraxial traced ray of the full system. This difference however exactly resembles the aberrations of the ray at the image plane.

2.3 Summary of the general theorem

To summarize the above mathematical treatment we have found a mathematical exact prove, that the total aberration of any ray within an optical system consisting of n surfaces, can be expressed as the sum of individual surface contributions Δ_i. Those individual surface contributions result from the differential rays at surface i (i.e. the difference between linear and real ray tracing across surface i), which are according to Eq. (7) linearly propagated to the image.

In other words with Eq. (11) we have found a mathematical relation between surface aberration contributions and the exact ray aberrations at the image plane, including all orders, quite similar to Aldis theory, however valid for freeform systems.

The big advantage of this method is that the individual surface contributions can exactly be identified, which is not possible with any other method. This allows the designer to identify strongly contributing surfaces, quite similar to a Seidel-aberration diagram. The individual contributions Δ_i can be obtained from ray-tracing data only, if additional dummy surface are defined. No assumptions are made about symmetry or surface shape.

3. Application example: freeform prism

In order to demonstrate the application of the general matrix theory presented above, a patented freeform prism [4], for head-mounted-displays, is used as a demonstrator. The basic design form is already depicted in Fig. 2. A detailed illustration of the design and ray-tracing is shown in Fig. 3 and the detailed design data are listed in Table 1 below.

Fig. 3 Freeform prism from reference [4].

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Table 1. Lens data of the freeform prism of reference [4] including additional dummy surfaces

View Table

What should be mentioned is that the tilt and decenter of these surfaces are all in global coordinates and referred to the entrance pupil (surface 0). The horizontal and the vertical field angle of this lens as we analyze here is +/−15° in each dimension. Besides, the pupil diameter is 4mm. The surfaces 1, 2 and 3 are anamorphic freeform surfaces, where K_x and K_y represent the conic constant, AR and BR describe the 4th and 6th order symmetric coefficients, AP and BP illustrate the 4th and 6th order asymmetric coefficients, Y and Z correspond to the decenter of the surface in Y and Z direction, and α shows the surface tilt.

3.1 Preparatory design modifications and calculation method

As a prerequisite for the described method, the design needs to be prepared for the necessary data extraction. First a reference ray (corresponding to the optical axis in a symmetric system) needs to be defined. We choose the same ray which is used in reference [4], i.e. the reference ray is the center ray on the first surface (pupil) starting with an angle of 0°. The reference ray is traced through the freeform system, defining surface intersection points at each surface i = 1…4. At each intersection we insert a pair of dummy surfaces D_i and D’_i, with their origin at the surface intersection point. The position of both dummy surfaces is identical to the reference ray intersection point, however the first dummy is sequentially before, whereas the second dummy is sequentially after the surface in a sequential raytracer. Moreover the surfaces are tilted to be exactly orthogonal to the reference ray. The position of these dummy surfaces is also listed in the above Table 1. Note that positioning these dummy surfaces can be performed after any global ray-tracing of the reference ray or by simple optimization of the positions and angles of the dummy surfaces. Further note that raytracing of arbitrary rays may correspond to non-physical raytracing at the dummy surfaces, meaning that the ray is traveling backwards. However this is not a problem, since we only use the intersection data for our mathematical treatment and sequential ray-tracers will perform this tracing correct.

As a further simplification to the design, which however is not mandatory, we removed the small image tilt of 3° relative to the reference ray and oriented the image surface strictly orthogonal to the reference ray. This step is not really required for our method, since as we are predicting ray-coordinates at some final orthogonal reference plane, we can always calculate the ray-intersection in an inclined image plane as well.

In a last preparation step we have to determine the 4-dimensional ABCD transformation matrices S_i and T_i. For the latter matrices T_i, according to Eq. (1), all that is required are the distances d_i between consecutive parallel dummy surfaces D’_i and D_{i + 1}. These data are available from any ray-tracer by reporting the optical path length along the reference ray. Determination of the linear transformation S_i of the tilted and decentered real surfaces is more challenging and two methods have been implemented: A set of differential rays around the reference ray can be traced to calculate their intersection coordinates and angles at the dummy surface pair D_i and D’_i. From this set of rays the linear matrix S_i, can be found by solving a sufficiently large set of linear equations. Alternatively we have used the predefined ABCD-function for tilted and decentered systems in Synopsis CodeV10.8 [24]. This function is able to calculate the four-dimensional ABCD-matrix between any surface pair, also for tilted and decentered systems. A comparison of both methods shows that the results are identical, so for the further analysis a CodeV macro was developed, providing all necessary ray-tracing, matrix calculations and data processing.

Following the above preconditioning, we have determined all necessary transformation matrices S_i, T_i and via Eq. (5) also M_I,i for the aberration calculation as described in section 2. Therefore calculation of the aberration contribution Δ_i of every ray at the image is now possible via Eq. (7), if in a prior real ray-tracing the ray-coordinates r_i and r’_i are recorded.

3.2 Distortion analysis results

As a first simple application the aberration contribution of a set of single rays can be studied. The chief-rays, representing different field angles, allow for a first useful application of the method. Since lateral aberrations of the chief-rays at the image plane correspond to the distortion of the system, we can simply apply our method to study distortion contributions in the freeform prism. To do so, we trace a set of chief-rays (i.e. rays starting at the center of the pupil) through the system. We use a set of 11x11 rays, scanning the field of view of +/−15° in the x and y-direction. For every ray the ray-coordinates r_i and r’_i are recorded at all dummy surfaces. By applying Eq. (7) and Eq. (3) we can calculate the aberration contribution vector Δ_i on the image plane. The spatial (x,y) components of this aberration vector correspond to lateral distortion, i.e. the distortion, which is induced by each of the surfaces. This is illustrated in Fig. 4.

Fig. 4 Distortion analysis: In (a) – (d) the surface contributions are shown, where the red circles are the paraxial positions of the rays on image, and the blue cross represents the lateral distortion contribution of the surface, (e) illustrates the sum of the distortion contributions and is compared to the exact ray-tracing result at the image, (f) shows the surface contributions for a single ray.

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In Figs. 4(a)-4(d), red circles are the ideal positions of the rays on the image, calculated by the linear system matrixes, whereas the blue cross represents the ray aberration contributions from surfaces 1 to 4 at the image plane, therefore the deviations of the blue cross from the red dots are the distortion contributions of each surface. The summation of all distortion contributions corresponds to the overall distortion at image surface, as drawn as blue crosses in Fig. 5(e) and compared to the red circles directly obtained from real ray-tracing of the system to the image in CodeV. As expected from Eq. (11) they exactly agree at the image plane. The additional information we get from this analysis, as compared to simple ray-tracing, are the surface resolved contributions of each surface, as depicted in Fig. 4(f), quite similar to a Seidel-aberration diagram in rotational-symmetric systems. For freeform systems the surface-resolved contributions are useful information for the designer to concentrate on large contribution surfaces during optimization.

Fig. 5 Transverse aberrations for the (0°,0°) field, resolved for individual surface contributions.

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3.4 Transverse aberration from ray fans

The method can now also be applied to calculate and illustrate transverse ray-aberrations for a particular spot in the image plane. As an example in Fig. 5 the meridional and saggital ray aberration contributions are calculated for the central spot (0°,0°).

The figure shows the meridional and saggital contributions of the individual surfaces to the overall transverse aberration at the image. The sum of all surfaces, plus the paraxial contribution, perfectly coincides with the results achieved by exact tracing of these rays to the image surface, which again proves the validity of Eq. (11).

4. Conclusions

This paper presents a matrix based method to calculate aberration contributions in freeform systems. It was mathematical proven, that the exact ray aberrations at the image plane is identical to the sum of individual surface contributions. Those individual surface contributions can be calculated form a ray-tracing analysis of the system and paraxial propagation of the real and paraxial differential vector induced from an individual surface. All calculations can be performed with standard ray-tracing, provided additional dummy surfaces are installed in the system before. No assumptions about symmetry were required. In other words the method resembles a kind of Aldis theory for freeform systems, allowing the exact calculation of surface aberration contributions.

Furthermore we have illustrated the method at the example of a freeform prism and shown that distortion and transverse ray-fans can be calculated and agree with the result of a standard ray-tracer.

This general work thus allows an “easy to implement” analysis of the aberration contributions within any freeform system, just like Seidel-diagrams. This will be beneficial in comparing freeform designs, tolerance analysis and improved optimization strategies. Extending this work to other aberration descriptions, such as Zernike-Polynomials is straightforward and will be subject of further work.

Acknowledgments

Thanks to the support by Chinese Scholarship Council and Stuttgart Research Center of Photonic Engineering.

References and links

1. K. P. Thompson and J. P. Rolland, “Freeform Optical Surfaces: A Revolution in Imaging Optical Design,” Opt. Photonics News 23(6), 30–35 (2012). [CrossRef]

2. O. Cakmakci and J. Rolland, “Head-Worn Displays: A Review,” J. Disp. Technol. 2(3), 199–216 (2006). [CrossRef]

3. T Gissibl, S Thiele, A Herkommer, and H Giessen, “Two-photon direct laser writing of ultracompact multi-lens objectives,” Nat. Photonics 10(8), 554–560 (2016). [CrossRef]

4. K. Takahashi, “Head or face mounted image display apparatus,” U.S. Patent No. 5,701,202. 23 Dec. 1997.

5. Q. Meng, W. Wang, H. Ma, and J. Dong, “Easy-aligned off-axis three-mirror system with wide field of view using freeform surface based on integration of primary and tertiary mirror,” Appl. Opt. 53(14), 3028–3034 (2014). [CrossRef] [PubMed]

6. A. Bauer and J. P. Rolland, “Design of a freeform electronic viewfinder coupled to aberration fields of freeform optics,” Opt. Express 23(22), 28141–28153 (2015). [CrossRef] [PubMed]

7. J. M. Rodgers, “Catoptric optical system including concave and convex reflectors,” (1994). US Patent 5,309,276.

8. 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]

9. A. Cox, A System of Optical Design (Focal Press, 1964).

10. S. H. Brewer, “Surface Contribution algorithms for analysis and optimization,” J. Opt. Soc. Am. 66(1), 8–13 (1976). [CrossRef]

11. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

12. R. Kingslake and R. B. Johnson, Lens Design Fundamental (SPIE Press, 2010).

13. W. Smith, Modern Optical Engineering (McGraw-Hill, 2000).

14. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005). [CrossRef] [PubMed]

15. K. Fuerschbach, J. P. Rolland, and K. P. Thompson, “Theory of aberration fields for general optical systems with freeform surfaces,” Opt. Express 22(22), 26585–26606 (2014). [CrossRef] [PubMed]

16. A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier, 2005).

17. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2010).

18. H. Gross, ed., Handbook of Optical Systems, Volume 1, Fundamentals of Technical Optics (Wiley-VCH, 2005)

19. D. Rausch and A. M. Herkommer, “Phase space approach to the use of integrator rods and optical arrays in illumination systems,” Adv. Opt. Technol. 1(1–2), 69–78 (2012).

20. J. Koshel, Illumination Engineering: Design with Nonimaging Optics (Wiley-IEEE, 2013).

21. A. M. Herkommer, “Phase space optics: an alternate approach to freeform optical systems,” Opt. Eng. 53(3), 031304 (2013). [CrossRef]

22. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69(12), 1710–1716 (1979). [CrossRef]

23. B. Chen and A. M. Herkommer, “High order surface aberration contributions from phase space analysis of differential rays,” Opt. Express 24(6), 5934–5945 (2016). [CrossRef] [PubMed]

24. Synopsis CodeV10, 8 News, https://optics.synopsys.com/codev/codev-whatsnew.html

Surface	Surface shape / radius (mm)		Surface position	Inclination angle(α)	Glass (n, v)
0	Infinity (pupil)		Origin
D1	Infinity		Z 30.002
1	R_y −108.187	AR 5.542E-7		−14.700°	1.492, 57.471
	R_x −73.105	BR 8.176E-11	Y −24.028
	K_y 0	AP −0.080	Z 26.360
	K_x 0	BP −1.379
D1’	Infinity		Z 30.002	−1.066 ˚	1.492, 57.471
D2	Infinity		Y −0.251 Z 43.485	−1.066 ˚	1.492, 57.471
2	R_y −69.871	AR-7.233E-11		36.660°	1.492, 57.471
	R_x −60.374	BR-4.529E-12	Y 19.109
	K_y −0.1368	AP 29.075	Z 33.339
	K_x −0.123	BP −2.085
D2’	Infinity		Y −0.251 Z 43.485	38.376 ˚	1.492, 57.471
D3	Infinity		Y −11.858 Z 28.827	38.376°	1.492, 57.471
3	R_y −108.187	AR 5.542E-7		−14.700°	1.492, 57.471
	R_x −73.105	BR 8.176E-11	Y −24.028
	K_y 0	AP −0.080	Z 26.360
	K_x 0	BP −1.379
D3′	Infinity		Y −11.858 Z 28.827	−55.019°	1.492, 57.471
D4	Infinity		Y −23.067 Z 36.667	−55.019°	1.492, 57.471
4	77.772		Y −35.215	−47.770°
			Z 18.817
D4’	Infinity		Y −23.067 Z 36.667	−50.668°
5(Image)	Infinity		Y −30.892	−50.668°
			Z 43.083

Generalized Aldis theorem for calculating aberration contributions in freeform systems

Abstract

1. Introduction

2. Phase space in non-rotational symmetric systems

2.1. Concept of phase space and definitions

2.2. Aberration calculation

2.3 Summary of the general theorem

3. Application example: freeform prism

3.1 Preparatory design modifications and calculation method

3.2 Distortion analysis results

3.4 Transverse aberration from ray fans

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Tables (1)

Equations (11)

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