Freeform aplanatic systems as a limiting case of SMS

Juan C. Miñano; Pablo Benítez; Bharathwaj Narasimhan

doi:10.1364/OE.24.013173

1. Introduction

Simultaneous Multiple Surface (SMS) design methods have been studied extensively in the past [1–3]. There has been no significant investigation into the link between the SMS method and aplanatic systems. We present an overview of the SMS design method as a solution to the coupling of three input wavefronts to three output wavefronts, giving a detailed mathematical insight into its limiting case when the input wavefronts approach each other and the output wavefronts do approach each other. We prove that an optical system obtained as a result of this scenario becomes aplanatic in the limiting case meaning that such a system forms a perfect image of the infinitesimal area around the point of interest.

Aplanatic systems with rotational symmetry are a result of optical systems corrected for spherical aberration, satisfying the Abbe sine condition. Two mirror rotationally symmetric aplanatic systems have already been extensively investigated in [4–7] and were primarily designed for telescopes which required superior aberration-free imaging characteristics. The surface profiles were analytically formulated by Schwarzschild, 1905 in [8] for rotationally symmetric systems and has been quite well known since. Recently Lynden-Bell et al. gave an analytic formulation of any two-mirror aplanatic design with rotational symmetry, which henceforth will be called 2D aplanatic since only the 2D cross section intervenes in those designs.

2. SMS method – an overview

SMS method was developed as a non-imaging optic design tool during the 1990s. It was initially developed for 2D cases by Miñano and later was extended for three dimensions by Benítez and Miñano.

SMS design method computes N optical surfaces used for focusing N input wavefronts onto N output wavefronts, as, for instance, the 2 plane input wavefronts normal to v₁ and v₂ shown in Fig. 1 which are focused onto 2 spherical wavefronts centered on points A₁ and A₂. This correspondence between the number of optical surfaces and that of wavefronts to couple is no longer consistent when the footprints of the design bundles do not occupy the full SMS surfaces, as demonstrated in [9]. These cases are not relevant for the purpose of this paper.

Fig. 1 Illustration of the SMS design principle in 2d and 3d cases.

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3. Aplanatism and its link to SMS design method

Ernst Karl Abbe demonstrated in 1873 [10] that rotational symmetric optical systems adhering to the sine condition were free of circular coma. Any offence against this resulted in aberrations which had linear dependence over the field angle. Many other recent studies have quantified this effect and have also derived the sine condition under more generalized circumstances [11–14]. The treatments presented in these papers involved the formulation of ray aberrations as a function of the offense against the sine condition. We provide an alternate treatment to the derivation of the sine condition under rotationally asymmetric prescription between the source and target using the étendue-conservation theorem [15]. This treatment does not assume any symmetry in the optical system and does not involve the computation of ray and or wave aberrations of the generalized system as shown below.

Consider a ray characterization by the Cartesian coordinates (x, y) when intersecting the plane z = 0 (object plane) and by the optical direction cosines (p, q) at this point of intersection, where (p, q, r) are optical direction cosines with respect to x, y and z axes respectively. The refractive indices of the medium in the object and image plane are denoted by n and n´ respectively. The optical direction cosines fulfill p² + q² + r² = n²(x,y,z). A similar characterization is used in the image plane x´, y´ as shown in Fig. 2. Let’s group the coordinates into 2 vectors, ρ and t, defined as follows:

Fig. 2 Nomenclature used in establishing the link between SMS and aplanatism.

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\begin{array}{l} ρ = (x, y, 0) & t = (p, q, r) \\ ρ^{'} = (x^{'}, y^{'}, 0) & t^{'} = (p^{'}, q^{'}, r^{'}) \end{array}

A two parametric bundle of rays can be defined by a couple of vector functions ρ(u,v) and t(u,v) where u and v are the 2 parameters of the bundle. This same set of rays can also be characterized at the image plane by another couple of functions ρ’(u,v) and t’(u,v). Let us recall now Herzberger’s Fundamental Optical Invariant [16–18]. This invariant establishes that ρ’_u⋅t’_v − ρ’_v⋅t’_u = ρ_u⋅t_v − ρ_v⋅t_u, for any two-parametric set of rays where u and v are the parameters (sub-indices denote partial derivative). This invariant is the same as the étendue conservation for 2-parametric bundles [15].

Now consider the particular case of a two-parametric set of rays where the parameters u and v coincide with x and p. Then ρ_x⋅t_p − ρ_p⋅t_x = 1. Choosing other couples of input coordinates as parameters we can get the following equations

\begin{matrix} {ρ^{'}}_{x} \cdot {t^{'}}_{p} - {ρ^{'}}_{p} \cdot {t^{'}}_{x} = ρ_{x} \cdot t_{p} - ρ_{p} \cdot t_{x} = 1 \\ {ρ^{'}}_{x} \cdot {t^{'}}_{q} - {ρ^{'}}_{q} \cdot {t^{'}}_{x} = ρ_{x} \cdot t_{q} - ρ_{q} \cdot t_{x} = 0 \\ {ρ^{'}}_{y} \cdot {t^{'}}_{q} - {ρ^{'}}_{q} \cdot {t^{'}}_{y} = ρ_{y} \cdot t_{q} - ρ_{q} \cdot t_{y} = 1 \\ {ρ^{'}}_{y} \cdot {t^{'}}_{p} - {ρ^{'}}_{p} \cdot {t^{'}}_{y} = ρ_{y} \cdot t_{p} - ρ_{p} \cdot t_{y} = 0 \end{matrix}

Let us now consider the image space coordinates x’, y’, p’, q’ as functions of the object space variables. Expanding the image coordinates in terms of x and y, we get (only terms until the first order in x or y have been explicitly written.):

\begin{array}{l} x ´ = A (x, y, p, q) = a_{00} (p, q) + a_{10} (p, q) x + a_{01} (p, q) y + ... \\ y ´ = B (x, y, p, q) = b_{00} (p, q) + b_{10} (p, q) x + b_{01} (p, q) y + ... \\ p ´ = C (x, y, p, q) = c_{00} (p, q) + c_{10} (p, q) x + c_{01} (p, q) y + ... \\ q ´ = D (x, y, p, q) = d_{00} (p, q) + d_{10} (p, q) x + d_{01} (p, q) y + ... \end{array}

One of the conditions required to achieve aplanatism is that all rays satisfy the stigmatic condition between one object and one image point. Coordinate systems are chosen so these points are the origins, so in mathematical terms this condition is just a₀₀ = b₀₀ = 0.

\begin{matrix} x^{'} = A (0, 0, p, q) = a_{00} (p, q) = 0 & y^{'} = B (0, 0, p, q) = b_{00} (p, q) = 0 \end{matrix}

Using Eqs. (2), (3) and Eq. (4) for the point x = y = 0 we get,

(\begin{matrix} a_{10} & b_{10} \\ a_{01} & b_{01} \end{matrix}) (\begin{matrix} c_{00 p} & c_{00 q} \\ d_{00 p} & d_{00 q} \end{matrix}) = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

The other condition of aplanatism, by definition, is first order sharp imaging around the origin which means that the 4 terms of the first matrix in Eq. (5) are independent of p and q, i.e., a₁₀ _p = a₁₀ _q = a₀₁ _p = a₀₁ _q = b_10,_p = b₁₀ _q = b₀₁ _p = b₀₁ _q = 0, (a₁₀ _p represents the derivative of a₁₀ with respect to p) i.e., a₁₀, a₀₁, b₁₀, b₀₁ must be constants. Additionally we can assume without loss of generality that the x, y plane is rotated around the z axis to get a₀₁ = b₁₀ = 0. The other constants are called lateral magnifications M along x and y axes respectively: a₁₀ = M_X, b₀₁ = M_Y. Thus, using Eq. (5) for the aplanatic designs we get:

(\begin{matrix} c_{00 p} & c_{00 q} \\ d_{00 p} & d_{00 q} \end{matrix}) = (\begin{matrix} M_{X}^{- 1} & 0 \\ 0 & M_{Y}^{- 1} \end{matrix})

from which:

c_{00} = {p^{'}}_{0} + p / M_{X}

and

d_{00} = {q^{'}}_{0} + q / M_{Y}

, which leads to the generalized Abbe sine condition which states that, in an aplanatic system, the rays linking the origin (x = y = 0) with the other origin (x’ = y' = 0) must also fulfill

\begin{matrix} p^{'} = {p^{'}}_{0} + \frac{p}{M_{X}} & q^{'} = {q^{'}}_{0} + \frac{q}{M_{Y}} \end{matrix}

where p´₀ and q´₀ are constants that represent the variables p´ and q´ corresponding to the ray originating from x = y = 0 with direction p = q = 0. We can now write again Eq. (3) with the expressions of the coefficients a₀₀, a₁₀, a₀₁, b₀₀, b₁₀, b₀₁, c₀₀, c₁₀, c₀₁, d₀₀, d₁₀, d₀₁ calculated in Eqs. (4)–(7), to get

\begin{matrix} x^{'} = M_{X} x + ... & y^{'} = M_{Y} y + ... \\ p^{'} = {p^{'}}_{0} + \frac{p}{M_{X}} + ... & q^{'} = {q^{'}}_{0} + \frac{q}{M_{Y}} + ... \end{matrix}

where again only the terms until the first order have been explicitly written. Terms in x and y of order 2 and higher are represented by the suspension points. The coefficients of those higher order terms are arbitrary functions of p, q.

3.1 A general mathematical description of the SMS method

Let us define an SMS design as that optical design which images stigmatically two or more points of the object space into corresponding points of the image space. Stigmatic imaging of one object point is achieved using Cartesian ovals. The general expression of the image coordinate x´ = A(x,y,p,q) for an SMS design stigmatically imaging 3 object points: α = (x_α, y_α), β = (x_β, y_β) and γ = (x_γ, y_γ) onto 3 image points: α´ = (x´_α, y´_α), β´ = (x´_β, y´_β) and γ´ = (x´_γ, y´_γ) can be written as:

x^{'} = A_{0} + A_{1} x + A_{2} y + F (x, y, p, q, x_{α}, y_{α}, x_{β}, y_{β}, x_{γ}, y_{γ})

The constants A₀, A₁, A₂ can be calculated as:

(\begin{matrix} A_{0} \\ A_{1} \\ A_{2} \end{matrix}) = {(\begin{matrix} 1 & x_{α} & y_{α} \\ 1 & x_{β} & y_{β} \\ 1 & x_{γ} & y_{γ} \end{matrix})}^{- 1} (\begin{matrix} {x^{'}}_{α} \\ {x^{'}}_{β} \\ {x^{'}}_{γ} \end{matrix})

F is an arbitrary continuous function nulling at the points α, β and γ . Its general expression is

\begin{array}{l} F (x, y, p, q, x_{α}, y_{α}, x_{β}, y_{β}, x_{γ}, y_{γ}) = \\ \sum_{i, j, k = {0, 1}} A_{i j k} {(x - x_{α})}^{i} {(y - y_{α})}^{1 - i} {(x - x_{β})}^{j} {(y - y_{β})}^{1 - j} {(x - x_{γ})}^{k} {(y - y_{γ})}^{1 - k} \end{array}

where A_ijk are arbitrary continuous functions of x, y, p, q taking finite values at the points α, β and γ (i,j,k are Boolean variables taking values 0 or 1, then the sum in Eq. (11) has 8 addends). Note that

\begin{array}{l} {\frac{\partial F}{\partial x} |}_{\begin{array}{l} x = x_{α} \\ y = y_{α} \end{array}} = \sum_{j, k = {0, 1}} A_{1 j k} {(x_{α} - x_{β})}^{j} {(y_{α} - y_{β})}^{1 - j} {(x_{α} - x_{γ})}^{k} {(y_{α} - y_{γ})}^{1 - k} \\ {\frac{\partial F}{\partial y} |}_{\begin{array}{l} x = x_{α} \\ y = y_{α} \end{array}} = \sum_{j, k = {0, 1}} A_{0 j k} {(x_{α} - x_{β})}^{j} {(y_{α} - y_{β})}^{1 - j} {(x_{α} - x_{γ})}^{k} {(y_{α} - y_{γ})}^{1 - k} \end{array}

The function y´ = B(x,y,p,q) can be written in a similar fashion as x´. Let us now examine the case when the three points α = (x_α, y_α), β = (x_β, y_β) and γ = (x_γ, y_γ) are coincident with the origin and x´ = 0 for this coincident point. In this case A₀ = 0, as seen from Eq. (10).

It is clear from Eq. (12) that the quantities ∂F/∂x and ∂F/∂y at the point α = (x_α, y_α) are 0 in this limit case. Then, according to Eq. (9):

\begin{matrix} {\frac{\partial x^{'}}{\partial x} |}_{\begin{array}{l} x = x_{α} \\ y = y_{α} \end{array}} = A_{1} & {\frac{\partial x^{'}}{\partial y} |}_{\begin{array}{l} x = x_{α} \\ y = y_{α} \end{array}} = A_{2} \end{matrix}

Consider now the case when (x_α, y_α) = (0,0), (x_β, y_β) = (Δ_x,0), (x_γ, y_γ) = (0, Δ_y) and (x_α´, y_α´) = (0,0), (x_β´, y_β´) = (Δ_x´,0), (x_γ´, y_γ´) = (0, Δ_y´) (Δ_x, Δ_y, Δ_x´ and Δ_y´ are constants). According to Eq. (10) we have

(\begin{matrix} A_{0} \\ A_{1} \\ A_{2} \end{matrix}) = {(\begin{matrix} 1 & 0 & 0 \\ 1 & Δ_{x} & 0 \\ 1 & 0 & Δ_{y} \end{matrix})}^{- 1} (\begin{matrix} 0 \\ Δ_{x} ´ \\ 0 \end{matrix}) = (\begin{matrix} 1 & 0 & 0 \\ - Δ_{x}^{- 1} & Δ_{x}^{- 1} & 0 \\ - Δ_{y}^{- 1} & 0 & Δ_{y}^{- 1} \end{matrix}) (\begin{matrix} 0 \\ Δ_{x} ´ \\ 0 \end{matrix}) = (\begin{matrix} 0 \\ {Δ^{'}}_{x} / Δ_{x} \\ 0 \end{matrix})

The coefficients corresponding to y´ can be found in a similar way to be:

(\begin{matrix} B_{0} \\ B_{1} \\ B_{2} \end{matrix}) = (\begin{matrix} 0 \\ 0 \\ {Δ^{'}}_{y} / Δ_{y} \end{matrix})

From Eqs. (9) and (14), and their equivalents in the y´ coordinate, we get that in this limit case, the expansion (in terms of x, y) of the functions x´ = A(x,y,p,q) and y´ = B(x,y,p,q) at the point x = y = 0 is (only terms until the first order are explicitly written)

\begin{array}{l} x^{'} = \frac{{Δ^{'}}_{x}}{Δ_{x}} x + ... & y' = \frac{{Δ^{'}}_{y}}{Δ_{y}} y + ... \end{array}

which when compared to the 2 first equations of Eq. (8) reveals that a three point SMS design at the limit (when Δ_x, Δ_y, Δ_x´ and Δ_y´ → 0) is an aplanatic one with M_X = (Δ_x´/Δ_x) and M_Y = (Δ_y´/Δ_y). Application of Eq. (16) to Eq. (5) can be used to check that this SMS design fulfills the generalized Abbe sine condition represented in Eq. (7).

From the preceding result we also conclude that, in general, an aplanatic freeform design must contain a minimum of three optical surfaces since the SMS design for three points needs three surfaces. Only in particular cases, such as those resulting in rotational symmetry, the aplanatic design requires just two.

One such instance of a freeform aplanatic system is shown in Fig. 3, more can be found in [19]. It is a system containing 3 reflective surfaces with M_X = 2, M_Y = 1, p´₀ = 0 and q´₀ = 0.86. We are planning to publish more details and analysis of the same in a follow up manuscript to be submitted soon.

Fig. 3 A three mirror freeform aplanatic system showing rays linking the origin of the object space with the origin of the image space.

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4. Conclusions

The general condition for aplanatism in freeform optical systems, as represented in Eq. (7), has been obtained in a simple way. We have also shown that an aplanatic design can be viewed as a 3-point SMS design in the limit case when the 3 points are coincident. This result gives an insight about how to design aplanatic freeforms which will be discussed in a future manuscript.

Acknowledgments

UPM authors thank the European Commission (Erasmus Mundus IDEAS, ADOPSYS: FP7-PELE-2013-ITN Grant Agreement No. 608082, NGCPV: FP7-ENERGY.2011.1.1 Grant Agreement No. 283798), the Spanish Ministries (OPTIVAR: TEC2014-56867-R, GUAKS: RTC-2014-2091-7, SUPERRESOLUCION: TEC2011-24019, PMEL: IPT-2011-1212-920000), UPM (Q090935C59) and the academic license for CodeV from Synopsys for the support given to the research activity of the UPM-Optics Engineering Group.

References and links

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15. J. C. Miñano, R. Mohedano, and P. Benítez, Nonimaging Optics. The Optics Encyclopedia (Wiley, 2015).

16. M. Herzberger, “On the fundamental optical invariant, the optical tetrality principle, and on the new development of gaussian optics based on this Law,” J. Opt. Soc. Am. 25(9), 295–304 (1935). [CrossRef]

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19. B. Narasimhan, P. Benitez, J. C. Miñano, J. Chaves, D. Grabovickic, M. Nikolic, and J. Infante, “Design of three freeform mirror aplanat,” Proc. SPIE 9579, 95790K (2015).

Freeform aplanatic systems as a limiting case of SMS

Abstract

1. Introduction

2. SMS method – an overview

3. Aplanatism and its link to SMS design method

3.1 A general mathematical description of the SMS method

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (3)

Equations (16)

Optics Express