Design of two spherical mirror unobscured relay telescopes using nodal aberration theory

Samuel Steven; Julie Bentley; Alfredo Dubra

doi:10.1364/OE.27.011205

1. Introduction

The reflective relay telescope was first reported shortly after the reflective telescope objective in the 17^th century, although it was not until the late 19^th century that it became widely adopted after the invention of film photography, the spectroscope, and the glass mirror [1–3]. The reflective finite conjugate relay followed with widespread adoption in the 20^th century, where the development of reflective microscopes was one of its early applications [4–6]. Since then, lithography, microscopy, spectroscopy, holography, astronomy, and high energy laser applications have all incorporated reflective relays, often in an unobscured form to improve resolution and throughput [7–17]. Furthermore, applications such as those using adaptive optics (AO), analog optical image processing, and high energy lasers often require a 4f or modified 4f relay (2f₁ + 2f₂), that is well corrected in both the image and pupil [16,18–21].

However, the design of unobscured reflective 4f and 2f₁ + 2f₂ relays has been challenging due to the lack of a unified wavefront aberration theory of rotationally symmetric and non-symmetric optical systems. Early mathematical theories of non-symmetrical systems centered on small perturbations of rotationally symmetric systems [22–24], closed-form expressions of ray paths such as the Coddington equations [25], real ray tracing schemes [26,27], and expansion of the imaging properties around a central ray [28–30]. This last approach has more recently been generalized and demonstrated in a variety of unobscured mirror systems [31–37]. Vector aberration theory, also known as Nodal Aberration Theory (NAT), originally suggested by Buchroeder [38,39] and later formalized by Thompson [40], is an extension of rotationally symmetric aberration theory that is mathematically and conceptually simpler than previous theories. Among other applications, NAT has been used to design freeform surfaces [41], as well as unobscured reflective systems [12,42–45]. However, analytical expressions for starting geometries of unobscured reflective systems have not yet been proposed.

Here, we use third-order NAT to describe the aberrations of unobscured relays formed by two tilted spherical mirrors, and analytically find optimal starting geometries. The method is demonstrated for infinite and finite conjugate 2f₁ + 2f₂ relays, as well as for a modified 2f₁ + 2f₂ form. Additionally, the sensitivity of the infinite conjugate 2f₁ + 2f₂ relay to collimation is discussed. Finally, a natural extension to the optimization method is proposed to deal with multiple conjugates simultaneously.

2. Background

2.1 Modified 4f relay

A single optical element can be used to create an image of (relay) an object (finite conjugate) or pupil (infinite conjugate). However, when single element relays are combined in series, as shown in Fig. 1 for the infinite conjugate, the required size of the optical elements can become substantial, as a result of diverging rays from previous relays (Fig. 1, top row). An alternative with potentially more moderate optical element sizes and slower (larger f/#) optical elements is the two-element 4f relay (Fig. 1, bottom row).

Fig. 1 First-order layout of two identical consecutive infinite conjugate 4f relays, formed by either a single optical element (top) or two elements (bottom) with equal optical path length. The red and blue rays are the chief and marginal rays, respectively. The largest system element(s) are depicted in green.

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The 4f relay consists of two identical elements with focal length f, separated by a distance 2f, providing unit magnification. There are two types of 4f relays: a finite conjugate relay and a pupil relay, termed here as type 1 and type 2, respectively. In both cases the object (type 1) or pupil (type 2) is placed a focal length away from the first element, assumed to have one common principal plane, such that the conjugate plane is located a focal length away from the principal plane of the second element, giving a total system length of 4f. In 4f systems the chief ray emerges collimated for a type 1 system, while for a type 2 system, the marginal ray emerges collimated. Ignoring the diffraction blur and wavefront aberrations, these configurations reproduce the input electric field amplitude and phase in the image plane.

A 4f system can be altered to a 2f₁ + 2f₂ setup, achieving identical properties except for a change in the magnification, $m = - f_{2} / f_{1}$ , of the object (type 1) or pupil (type 2) (Fig. 2, top row). An additional modification can be made by changing the distance between the object (type 1) or pupil (type 2) and the first element such that the distance between the second element and the conjugate plane moves accordingly to retain conjugation. In such a modified 2f₁ + 2f₂ relay (Fig. 2, bottom row), image telecentricity for type 1 and collimation for type 2 relays are maintained, but the marginal and chief rays, respectively, between the two elements are no longer collimated. Despite the change in distance to the first element, image magnification remains the same for both relay types (see Appendix A for demonstration).

Fig. 2 First-order layout of a 2f₁ + 2f₂ relay (top) and a modified 2f₁ + 2f₂ (bottom) relay for type 1 (left) and type 2 (right). Note that the difference between the two forms is that distance between the object (type 1) or the pupil (type 2) to the first and last elements are different. The dashed black line represents the optical axis, the red lines the chief rays, and the blue lines the marginal rays.

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In what follows, we study modified 2f₁ + 2f₂ type 1(finite conjugate; Fig. 2, left) and type 2 setups (infinite conjugate; Fig. 2, right). To simplify notation, the marginal and chief ray of type 1 are set as the chief and marginal rays, respectively, of type 2. Both relay types have an input height (entrance pupil or object) $h_{0}$ , input angle $α_{0}$ (half field of view or numerical aperture), and a telecentric object space or a collimated input beam, respectively.

2.2 Unobscured two-mirror relay

Let us now introduce the notation to describe an unobscured modified 2f₁ + 2f₂ relay formed by two tilted spherical mirrors. The tilts of each surface are around the x- and y-axis that define the plane transverse to the optical axis ray (OAR) impinging on the mirror, shown in Fig. 3, right. Without loss of generality, the first mirror can be assumed to be tilted only with respect to the x-axis ( $θ_{y_{1}} > 0$ ), and the second mirror around both the x- and y-axis ( $θ_{y_{2}}$ , $θ_{x_{2}}$ ). As shown in Fig. 3, the OAR of such unobscured relays is defined as the ray that connects the center of the object with the center of a circular aperture [46].

Fig. 3 Top view of unobscured modified 2f₁ + 2f₂ relays formed by two tilted spherical mirrors of type 1 (left) and type 2 (middle), along with the system of coordinates as it propagates through the system (right). Note that although in the figure the second mirror is only tilted around the x-axis, it could also be tilted in the y-axis, such that the optical axis leaves the plane of the page. The dashed black line represents the optical axis ray (OAR), the red lines the chief rays, and the blue lines the marginal rays. Note that the coordinate system follows the OAR, except the z-axis does not flip sign upon reflection.

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In these relays mechanical clearance at the object (type 1) or entrance pupil (type 2) plane and the image plane (type 1) or exit pupil (type 2) are important in order to avoid vignetting of the area crossed by light rays at the center of the relay. For both type 1 and type 2 relays (see Fig. 4) with small mirror tilts ( $θ_{y_{1}}$ , $θ_{x_{2}}$ , $θ_{y_{2}}$ << 1) these clearances can be approximated as

δ_{1} \approx 2 | θ_{y_{1}} (f_{1} + ε) | - | h_{0} | - | \frac{α_{0} (f_{1}^{2} - ε^{2})}{f_{1}} |,

δ_{2} \approx 2 | m θ_{2} (f_{1} + m ε) | - | m h_{0} | - | \frac{α_{0} (f_{1}^{2} - m^{2} ε^{2})}{f_{1}} |,

where the object distance to the first spherical mirror d₁ is expressed in terms of its departure from the on-axis mirror’s focal length (

d_{1} = f_{1} + ε

) and

θ_{2} = \sqrt{θ_{x_{2}}^{2} + θ_{y_{2}}^{2}}

. These formulae show that increasing the focal lengths of the mirrors does not necessarily increase the clearances, although increasing the mirrors tilt does.

Fig. 4 Unobscured reflective 2f₁+ 2f₂ type 1 relay schematic showing mechanical clearances $δ_{1}$ and $δ_{2}$ .

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3. Third order aberration theory

3.1 Rotationally symmetric optical systems

Aberration theory studies the deviation of a wavefront converging towards an image plane from an ideal spherical shell. The wavefront of an optical system is usually described in reference to the intersection of individual light rays with both the exit pupil and image planes, in coordinates normalized to the pupil radius and the field of view, respectively, (see Fig. 5). The wavefront aberration of a rotationally symmetrical system can be expressed as a polynomial expansion calculated as the sum of each surface’s contribution in the optical system, each of which is an infinite series of weighted terms with field and pupil dependence [47],

W (H, θ = 0, ρ, φ) = \sum_{j} \sum_{p = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{m - 0}^{\infty} {(W_{k l m})}_{j} H^{k} ρ^{l} \cos^{m} φ,

where θ is chosen to be zero without loss of generality due to the rotational symmetry, k = 2p + m, l = 2n + m, and j denotes the surface (j, p, n, and m are positive integers). The W_klm coefficients for which k + l = 4 (their gradient is of 3rd order) are known as Seidel coefficients and can be calculated using paraxial ray tracing [48,49]. The wavefront can also be expressed in terms of vectors in a complex plane such that

\overset{⇀}{H} = H e^{i θ}

and

\overset{⇀}{ρ} = ρ e^{i φ}

,

W (\overset{⇀}{H}, \overset{⇀}{ρ}) = \sum_{j} \sum_{p = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{m - 0}^{\infty} {(W_{k l m})}_{j} {(\overset{⇀}{H} \cdot \overset{⇀}{H})}^{p} {(\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})}^{n} {(\overset{⇀}{H} \cdot \overset{⇀}{ρ})}^{m},

with the dot product defined as

\overset{⇀}{A} \cdot \overset{⇀}{B} = Re [\overset{⇀}{A}] Re [\overset{⇀}{B}] + Im [\overset{⇀}{A}] Im [\overset{⇀}{B}]

, (see [50]).

Fig. 5 Naming convention used to describe wavefront aberrations in terms of normalized coordinates in the pupil (ρ, φ) and field (H, θ) planes for rotationally symmetric optical systems.

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3.2 Non-rotationally symmetric optical systems

The extension of wavefront aberration theory to non-rotationally symmetric optical systems is not trivial. Buchroeder [39] introduced a vector to represent the shift of the center of the aberration field due to a surface tilt or decentration, a concept later refined by Thompson [40]. In this formalism, the point at which each aberration is zero is referred to as a node. In rotationally symmetric optical systems, the aberrations have either zero (e.g., spherical) or one on-axis node (e.g., coma and astigmatism). Each tilted or decentered surface $j$ has its aberration field shifted by a vector ${\overset{⇀}{σ}}_{j}$ so that $\overset{⇀}{H}$ in Eq. (4) is replaced with $\overset{⇀}{H} - {\overset{⇀}{σ}}_{j}$ , and thus the wavefront aberration is given by

W (\overset{⇀}{H}, \overset{⇀}{ρ}) = \sum_{j} \sum_{p = 0}^{\infty} \sum_{n = 0}^{\infty} \sum_{m - 0}^{\infty} {(W_{k l m})}_{j} {[(\overset{⇀}{H} - {\overset{⇀}{σ}}_{j}) \cdot (\overset{⇀}{H} - {\overset{⇀}{σ}}_{j})]}^{p} {(\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})}^{n} {[(\overset{⇀}{H} - {\overset{⇀}{σ}}_{j}) \cdot \overset{⇀}{ρ}]}^{m},

where the coefficients W_klm remain those of rotationally symmetric systems [50]. The result of this substitution is that each aberration in a non-rotationally symmetric system has as many nodes as its order in the field (H). Therefore, when considering only third-order aberrations (ignoring distortion) the maximum number of nodes an aberration can have is two, which is the case for astigmatism (2nd-order in field). When the nodes overlap, we say that they are degenerate.

Thompson [50] discussed in detail the effect of the shift ${\overset{⇀}{σ}}_{j}$ on third-order aberrations, which for convenience are defined and illustrated in Table 1, below. Some points worth noting are listed next. For each aberration, the location of the node(s) at the image is given by the sum of the aberration field shift of each surface, weighted by their respective aberration contribution (see Table 1). Spherical aberration, which does not change across the field in rotationally symmetric systems, remains constant in non-rotationally symmetric systems. Coma however, which has a linear dependence in the field of view in rotationally symmetric systems, retains this dependency, but with its node shifted. This is equivalent to adding a constant amount of coma ( ${\overset{⇀}{A}}_{131}$ ) across the field. Astigmatism and the medial focus surface have constant shifts in the field, respectively ${\overset{⇀}{B}}_{222}^{2}$ and ${\overset{⇀}{A}}_{220_{M}}$ , but due to its quadratic dependence with field, astigmatism has two nodes that are described by its constant field-shift and a linear field dependence ( ${\overset{⇀}{A}}_{222}$ ). Additionally, the medial focus surface has a longitudinal (axial) shift, given by $B_{220_{M}}$ , that is of the same form as defocus. Third-order distortion is not considered here because it does not result in image blur.

Table 1. Third-Order Aberrations (Excluding Distortion) in a Non-Rotationally Symmetric Optical System According to Nodal Aberration Theory^a

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3.3 Field shift calculation for tilted spherical mirrors

The field shift ${\overset{⇀}{σ}}_{j}$ is a normalized vector contained in the image plane and given by [50],

{\overset{⇀}{σ}}_{j} = - \frac{{\overset{⇀}{i}}_{j}^{*}}{i_{j}},

where

{\overset{⇀}{i}}_{j}^{*}

is calculated as [46]

{\overset{⇀}{i}}_{j}^{*} = {\overset{⇀}{N}}_{j} \times ({\overset{⇀}{R}}_{j} \times {\overset{⇀}{S}}_{j}),

with

{\overset{⇀}{N}}_{j}

being a unit vector normal to the object plane (pointing away from the direction of propagation),

{\overset{⇀}{R}}_{j}

is a unit vector along the OAR incident on the surface (also pointing away from the direction of propagation), and

{\overset{⇀}{S}}_{j}

is a unit vector normal to the surface at the intersection with the OAR, where for a mirror, this vector points away from the mirror surface. The normalization factor i_j is the angle between the unperturbed (i.e. before tilt and/or decenter) chief ray (for a point at the edge of the field of view) and the local surface normal. All three vectors in Eq. (5) have real x, y, and z components. Eventually, when using the vector

{\overset{⇀}{σ}}_{j}

for calculating the wavefront aberrations using Eq. (3), it must be converted to a 2-dimensional complex vector.

Let us now use these definitions to calculate the field shift ${\overset{⇀}{σ}}_{j}$ for a spherical mirror tilted around the x-axis shown in Fig. 6. In this case, we have ${\overset{⇀}{S}}_{j} = n_{j} [\begin{matrix} 0 & \sin θ_{y_{j}} & \cos θ_{y_{j}} \end{matrix}]$ , where n_j is the index of refraction of the mirror ( $n_{j} = {(- 1)}^{j}$ with n₀ the refractive index before the first mirror). In this way, ${\overset{⇀}{i}}_{j}^{*} = [\begin{matrix} 0 & - \sin θ_{y_{j}} & 0 \end{matrix}]$ . This result shows that for a tilted spherical mirror the shift in the aberration field of the surface happens along the x-axis and for small tilts $(\sin θ_{y_{j}} \sim θ_{y_{j}})$ by an amount proportional to the mirror tilt angle. To calculate the field shifts for subsequent surfaces, the OAR has to go through the center of the aperture stop and image center. It is important to note that the system of coordinates are relative to the OAR, and thus, it changes after each reflection.

Fig. 6 Sample geometry used for illustrating the field displacement vector calculation for a tilted spherical mirror. OAR stands for optical axis ray, AS denotes the aperture stop, C.C. is the mirror center of curvature, $\overset{⇀}{N}$ the object normal, $\overset{⇀}{R}$ the incident ray, $\overset{⇀}{S}$ the surface normal, and ${\overset{⇀}{σ}}_{j}$ is the field shift vector after the tilting and propagation of the coordinate system.

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3.4 Wavefront variance as a performance metric

The wavefront variance W_var at a field point $\overset{⇀}{H}$ over a (circular) pupil is defined as [50],

W_{var} (\overset{⇀}{H}) = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {[W (\overset{⇀}{H}, \overset{⇀}{ρ})]}^{2} ρ d ρ d φ - {(\frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} W (\overset{⇀}{H}, \overset{⇀}{ρ}) ρ d ρ d φ)}^{2} .

The field-averaged wavefront variance can be used as a metric that corresponds to the average behavior of aberrations over the entire field of view [51],

{\bar{W_{var}}}_{_{\overset{⇀}{H}}} = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} W_{var} (\overset{⇀}{H}) H d H d θ .

A discrete version of this metric can be found in commercially available optical design optimization software packages. OpticStudio (Zemax, Kirkland, WA, USA) includes root mean square (RMS) wavefront as one of their default merit functions. The metric returned by such a function is calculated using operands that evaluate the optical path difference (with respect to the mean) at a user-defined number of points over the pupil using either rectangular grid sampling or Gaussian quadrature sampling, with the latter providing equal area weighting. These optical path differences are evaluated for each field point and then the weighted sum of the values squared yields the square of the merit function,

M F^{2} = \frac{\sum_{j} W_{i} {(T_{i} - V_{i})}^{2}}{\sum_{j} W_{i}},

where W_i is the weight of the operand, T_i is the target value, and V_i is the evaluated value (Zemax OpticStudio 18.7 User Manual, November 2018). If equal area pupil sampling and field sampling are selected for evaluation the merit function is, effectively, a discrete field-averaged RMS wavefront.

To analytically study the unobscured reflective relay at both the finite and infinite conjugates, we evaluate this metric accounting for third-order aberrations. Defocus (W₂₀) and tilt (W₁₁) are also included because they can be used to minimize the RMS wavefront. Piston and distortion terms are however, excluded from the analysis because they do not result in image blur. The wavefront aberration is thus given by

\begin{array}{l} W (\overset{⇀}{H}, \overset{⇀}{ρ}) \approx W_{20} (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) + W_{11} (\overset{⇀}{H} \cdot \overset{⇀}{ρ}) + W_{040} {(\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})}^{2} + [(W_{131} \overset{⇀}{H} - {\overset{⇀}{A}}_{131}) \cdot \overset{⇀}{ρ}] (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) \\ + [W_{220_{M}} (\overset{⇀}{H} \cdot \overset{⇀}{H}) - 2 (\overset{⇀}{H} \cdot {\overset{⇀}{A}}_{220_{M}}) + B_{220_{M}}] (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) \\ + \frac{1}{2} [W_{222} {\overset{⇀}{H}}^{2} - 2 \overset{⇀}{H} {\overset{⇀}{A}}_{222} + {\overset{⇀}{B}}_{222}^{2}] \cdot {\overset{⇀}{ρ}}^{2}, \end{array}

which when substituted in Eq. (9) yields

\begin{array}{l} {\bar{W_{var}}}_{_{\overset{⇀}{H}}} = \frac{1}{720} [90 W_{11}^{2} + 60 W_{20}^{2} + 64 W_{040}^{2} + 45 W_{131}^{2} + 20 W_{220_{M}}^{2} + 10 W_{222}^{2} + 60 W_{20} W_{220_{M}} \\ + 120 W_{11} W_{131} + 60 W_{040} (2 W_{20} + W_{220_{M}}) + 60 B_{220_{M}} (2 W_{20} + 2 W_{040} + W_{220_{M}}) \\ + 60 B_{220_{M}}^{2} + 90 A_{131}^{2} + 60 A_{222}^{2} + 60 A_{220_{M}}^{2} + 30 B_{222}^{4} . \end{array}

If one notes that the

B_{220_{M}}

term (see Table 1) corresponds to a constant amount of defocus that can be corrected for, and thus ignored, then all the coefficients of terms that result from non-rotational symmetry (

A_{131}^{2}

,

A_{222}^{2}

,

A_{220_{M}}^{2}

, and

B_{222}^{4}

), termed here as the NAT coefficients, are all positive. Therefore, a non-rotationally symmetric system cannot perform better than its rotationally symmetric version. Also, and again ignoring the

B_{220_{M}}

term, there are no terms containing both Seidel and NAT coefficients. Thus, tilting or decentering elements will only affect the terms with NAT coefficients, which leads us to propose considering only the non-rotationally symmetric (NRS) terms for optimization. This simplified field averaged metric, termed here as the NRS field-averaged wavefront variance, is therefore

{{\bar{W_{var}}}_{_{\overset{⇀}{H}}}}_{N R S} = \frac{1}{24} [3 A_{131}^{2} + 2 A_{222}^{2} + 2 A_{220_{M}}^{2} + B_{222}^{4}] .

Finding tilts and decenters that cancel all the terms in Eq. (13) will result in a performance that is the same as its rotationally symmetric version (ignoring distortion).

To validate the use of third-order nodal aberration theory we compared the RMS wavefront derived from NAT against real ray tracing, for which we used OpticStudio. Two reflective relays with different magnifications and tilt angles (specifications in Table 2) were chosen to allow for mechanical clearance, as well as to consider planar and orthogonal OAR folding. For both type 1 and type 2 relays, the theoretical RMS wavefront was calculated by substituting Eq. (12) into Eq. (8), and is compared to the RMS wavefront in OpticStudio. Both RMS wavefronts were evaluated over a square 5 mm (type 1) and 5° (type 2) field of view, and their respective RMS wavefront magnitudes as a function of field position (Fig. 7, columns 1 & 2) for 500 nm light. The OpticStudio field map was simulated using 200 x 200 field sampling, a ray density of 20, and the chief ray as reference. Additionally, the magnitude of the normalized relative error $σ = (W_{R M S, N A T} - W_{R M S, O S}) / \max (W_{R M S, N A T})$ is plotted in Fig. 7 column 3. For both systems the relative difference was less than 0.5% over the field of view for type 1 and type 2 relays, due to the higher order aberrations accounted for by the ray tracing software.

Table 2. Design Parameters for Two Sample Unobstructed 2f₁ + 2f₂ Type 1 and 2 Relays Formed by Two Spherical Mirrors with a Focal Length of the First Mirror f₁ of 500 mm and Entrance Pupil Diameter of 5 mm^a

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Fig. 7 RMS wavefront error field maps as calculated by third-order nodal aberration theory (column 1), by real ray tracing (column 2), and relative error (see text for details) as compared to third-order nodal aberration theory prediction (column 3) for two unobscured reflective relay telescopes of type 1 and 2 (rows 1-4) over a normalized 5 × 5° and 5 × 5 mm field angle, respectively, with specification parameters listed in Table 2, and evaluated for 500 nm light.

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4. Two spherical mirror unobscured 2f₁ + 2f₂ relay

The simplicity of the unobstructed 2f₁ + 2f₂ reflective relay allows the use of paraxial ray tracing and NAT to analytically find starting configurations that minimize the field average RMS wavefront. In this process, magnification m, input height h₀, input angle α₀ and tilt $θ_{y_{1}}$ of the first mirror are inputs, while the tilt $θ_{y_{2}}$ and $θ_{x_{2}}$ of the second mirror, the distance d₁ to the first mirror, and the focal length of the first mirror are the outputs. The first step of the design process, which is summarized in Fig. 8, is the calculation of the Seidel coefficients. These are then combined with the calculated field shifts $\overset{⇀}{σ}$ introduced by each element, to get an expression for the field-averaged wavefront variance. The resulting expression, see Eq. (12), or its non-rotationally symmetric component, see Eq. (13), can then be analytically minimized using symbolic calculators such as Mathematica (Wolfram Research, Champaign, IL, USA) or Matlab (Mathworks, Natik, MA, USA). The resulting parameters can then be provided as input to ray tracing software for further optimization, accounting for higher order aberrations.

Fig. 8 Proposed optimization process for designing unobscured spherical reflective systems.

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4.1 The 2f₁ + 2f₂ relay

The Seidel coefficients for a 2f₁ + 2f₂ system formed by two spherical mirrors are shown in Table 3, for both type 1 and type 2 relays. In the type 2 relay coma (W₁₃₁) is zero, irrespective of magnification. Otherwise, since m must be negative to have a real conjugate plane, the only other aberration in either relay that can be cancelled is coma in the type 1 relay, provided 1:1 magnification is possible. This is important, as W₁₃₁ is the only Seidel coefficient that is independent of the first mirror’s focal length. In the type 2 relay, all coefficients are inversely proportional to an order of the focal length of the first mirror, indicating that increasing the relay length could be used to mitigate these aberrations. For the type 1 relay, however, only astigmatism and medial focus curvature have inversely proportional relationships with the focal length of the first mirror, while spherical aberration has a directly proportional relationship with focal length.

Table 3. Expressions for Seidel Coefficients for Unobscured Two-Mirror 2f₁ + 2f₂ Type 1 and 2 Relays^a

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The NAT coefficients, calculated using the field shift vectors are shown below in Table 4. Notably, for this optical system, $W_{222}$ and $W_{220_{M}}$ have the same absolute value, and therefore $A_{222}^{2}$ and $A_{220_{M}}^{2}$ are equal (see definitions in Table 1). As with the Seidel coefficients, for the type 2 relay, the nodal aberration coefficients are all inversely proportional to a power of the focal length of the first mirror. Also, $A_{222}^{2}$ ( $A_{220_{M}}^{2}$ ) are not dependent on magnification. In the type 1 relay however, only $A_{131}^{2}$ and $B_{222}^{4}$ are proportional to an order of the focal length and all the coefficients depend on the relay’s magnification. Finally, and most importantly, the formulae show that some of the coefficients could be cancelled with appropriate selection of the mirror tilts, although simultaneous correction of all terms in one of the conjugates is not possible for negative magnifications. It is noted that ${\overset{⇀}{σ}}_{1}$ and ${\overset{⇀}{σ}}_{2}$ are proportional to $2 f_{1} / h_{0}$ and $2 / α_{0}$ for the type 1 and 2 relays, respectively.

Table 4. Surface $\overset{⇀}{σ}$ and Coefficients of Non-Rotationally Symmetric Field-Averaged Wavefront Variance for an Unobscured Two-Mirror Telescope

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The NAT coefficients can be used to calculate the third-order expression for the NRS field-averaged wavefront variance using Eq. (13), yielding

\begin{array}{l} {}_{T 1}{{\bar{W_{var}}}_{_{\overset{⇀}{H}}}}_{N R S} = \frac{α_{0}^{4}}{384 m^{4}} {4 m^{2} h_{0}^{2} [{(m θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}] + 3 f_{1}^{2} α_{0}^{2} [{(m^{2} θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}] \\ + 4 m^{2} f_{1}^{2} [{(m θ_{y_{1}}^{2} - θ_{y_{2}}^{2} + θ_{x_{2}}^{2})}^{2} + 4 θ_{x_{2}}^{2} θ_{y_{2}}^{2}]} \end{array}

for the type 1 relay and

\begin{array}{l} {}_{T 2}{{\bar{W_{var}}}_{_{\overset{⇀}{H}}}}_{N R S} = \frac{h_{0}^{4}}{384 f_{1}^{4}} {4 f_{1}^{2} α_{0}^{2} [{(θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}] + 3 h_{0}^{2} [{(θ_{y_{1}} + m θ_{y_{2}})}^{2} + m^{2} θ_{x_{2}}^{2}] \\ + 4 f_{1}^{2} [{(θ_{y_{1}}^{2} - m θ_{y_{2}}^{2} + m θ_{x_{2}}^{2})}^{2} + 4 m^{2} θ_{x_{2}}^{2} θ_{y_{2}}^{2}]} \end{array}

for the type 2 relay. The local extrema of these metrics for a given

θ_{y_{1}}

can be found analytically as the roots of the system of equations formed by their first derivatives. Because the field average variances are fourth-order polynomials in the second mirror x- and y-tilts, there are three pairs of x- and y-tilt angles that minimize each variance (Appendix B). The first two solutions are symmetric around the

θ_{x_{2}}

axis, while the third solution lies along the

θ_{y_{2}}

axis. Therefore, the first two solutions correspond to a 3-dimensional folding of the OAR, and in the third solution, the OAR is contained within a plane.

4.2 Example designs

To illustrate how the analytical solutions from the previous sections compare to those obtained by numerical ray tracing, three sample relays with parameters listed in Table 5 were evaluated using ray tracing software. The resulting NRS field-averaged wavefront variances as a function of the tilt angles of the second mirror, Eq. (14) and Eq. (15), are plotted below in Fig. 9, and compared to the analytical minima, see Appendix B, and also listed in Table 5 as coordinates $(θ_{x_{2}}^{N A T}, θ_{y_{2}}^{N A T})$ . The angles found through optimization using OpticStudio with 16 field points of equal area weighting and only $θ_{y_{2}}$ and $θ_{x_{2}}$ as variables, are also shown in the plot. The merit function was set up using RMS wavefront error, referenced to the chief ray, with 10 rings and 10 arms defining the pupil integration (Gaussian quadrature) sampling for the default merit function. The average difference between the angles found analytically and using OpticStudio for both conjugates over all three designs is 0.03° (range 0.0001 - 0.13°).

Table 5. Design Parameters of Sample Relays^a

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Fig. 9 Contour plots of the NRS field-averaged wavefront variance as a function of mirror two tilt angle for type 1 (top) and type 2 (bottom) relays for three designs (rows 1-3) showing predicted (green cross) and actual angles (red circle) for minimum field-averaged wavefront variance. Each contour plot shows 20 levels with step sizes 0.006, 7x10⁻⁶, and 5x10⁻⁵ μm² for type 1 designs 1-3, respectively, and 2x10⁻⁵, 0.02, and 0.5 μm² for type 2 designs 1-3, respectively.

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Next, we impose the mechanical considerations for mirror two as expressed by Eq. (2). The minimal clearance to avoid vignetting requires that $δ_{2} \geq 0$ , however, to allow for mounting we set the clearance $δ_{2} = 20$ mm. To illustrate this constraint, the clearance is plotted as a transparent red circle on top of the total field-averaged wavefront variance from design 1 of Fig. 9, shown below in Fig. 10. The circle represents all the tilt angles of mirror 2 that do not meet the mechanical constraint criteria. In the type 2 relay, the angles that achieve minimum field-averaged wavefront variance are outside the mechanical clearance constraint, however, in the type 1 relay such angles are not. Thus, to find the analytical minima given mechanical considerations the constraints on the angles must be employed. For the current example this results in tilt angles for mirror two of ( ± 3.52°, −0.11°). This is equivalent to introducing a constraint on the magnitude of the tilt angle of mirror two during optimization in real ray tracing design software.

Fig. 10 Infinite (left) and finite (right) conjugate field-averaged wavefront variance of design 1 as a function of mirror two tilt angle showing minimal required angle for a mechanical mounting clearance of 20 mm. The predicted (green cross) and actual angles (red circle) for minimum wavefront are also included.

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4.3 In-plane and orthogonal geometries

Two types of geometries have been preferentially used for unobscured reflective two-mirror relays. First, the in-plane geometry (Fig. 11, left) in which the OAR is contained within a plane such that $θ_{x_{2}} = 0$ [12,13,52–58], and second, the orthogonal geometry (Fig. 11, right) in which the plane containing the OAR after the second mirror is perpendicular to the plane containing the OAR before and after the first mirror such that $θ_{y_{2}} = 0$ [16,55,59,60].

Fig. 11 Top view of an unobscured reflective modified 2f₁ + 2f₂ type 1 relay with the tilt of the second element such that it has an in-plane geometry (left) and an orthogonal geometry (right). The dashed black line represents the optical axis ray (OAR), the red lines the chief rays, and the blue lines the marginal rays.

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Investigation of Table 4 highlights the advantages and disadvantages of the in-plane and orthogonal geometries. For the orthogonal geometry, in both relay types only $B_{222}^{4}$ can be canceled with the appropriate choice of tilt, irrespective of sign. All other coefficients, however, are proportional to the square sums of the two tilt angles and thus only by reducing the tilt angles can the aberration coefficient be reduced. Contrarily, in the in-plane geometry and for both relay types, all coefficients except for $B_{222}^{4}$ can be canceled with the appropriate choice of tilt, although the relative sign of tilt angles is opposite for $A_{131}^{2}$ and $A_{222}^{2}$ , $A_{220_{M}}^{2}$ such that they cannot be simultaneously canceled. The coefficient $B_{222}^{4}$ can mitigated only by reducing the tilt. This indicates that when $B_{222}^{4} ≫ A_{131}^{2}, A_{222}^{2}, A_{220_{M}}^{2}$ , an orthogonal geometry should be considered, while if $B_{222}^{4} ≪ A_{131}^{2}, A_{222}^{2}, A_{220_{M}}^{2}$ , an in-plane geometry should be considered instead. Geometries between orthogonal and in-plane should be considered if neither of these conditions are met. For an orthogonal geometry, constant astigmatism cancels out when $θ_{x_{2}} = \pm | θ_{y_{1}} | \sqrt{- m}$ for type 1 relays or $θ_{x_{2}} = \pm | θ_{y_{1}} | / \sqrt{- m}$ for type 2 relays, in agreement with the predictions from the Coddington equations [60]. Furthermore, this highlights a potential benefit for such orthogonal geometries, as the cancellation of constant astigmatism is independent of the sign of the angle leading to high flexibility when combining such relay with other optical systems.

Two examples of unobscured reflective type 2 relays are considered next both with in-plane and orthogonal folding (parameters listed in Table 6). The magnitude of the tilts were chosen to have mechanical clearances of $δ_{1}, δ_{2} > 20$ mm. The NAT coefficients for each design and geometry are shown in the bar plot below in Fig. 12. In all designs and conjugates constant astigmatism, $B_{222}^{2}$ , had the largest magnitude and in design 1 was minimized or canceled in the orthogonal geometry for both conjugates. Constant astigmatism became the dominant aberration for the orthogonal design. However, for the chosen design parameters, the in-plane configuration offers better correction for all the other coefficients as compared to the orthogonal configuration. Importantly, the magnitude of the aberrations of design 2 are over ten times the magnitude of the aberrations in design 1 as a result of the magnification and magnitude of the tilt of mirror two.

Table 6. Design Parameters for Two Infinite Conjugate 2f₁ + 2f₂ Relay Examples Formed by Two Spherical Mirrors (f₁ = 500 mm, h₀ = 4 mm and α₀ = 1.0°)^a

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Fig. 12 Magnitude of non-rotationally symmetric coefficients from field-averaged wavefront variance for designs 1 and 2 (rows 1 and 2) both in-plane and orthogonal geometries, for type 1 and type 2 relays (columns 1 and 2). Note the difference in scales between designs 1 and 2.

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4.4 Modified 2f₁ + 2f₂ relay

The NAT coefficients for the reflective 2f₁ + 2f₂ relay (Table 4) indicate that the tilting of the second spherical mirror in the telescope provides very limited aberration mitigation. The additional degree of freedom of varying d₁ provided by the modified 2f₁ + 2f₂ relay could be used to achieve improved aberration correction. The Seidel coefficients and their roots for both type 1 and 2 relays are shown below in Table 7 and Table 8, and are illustrated in Fig. 13 for three generic systems of magnifications m = −0.5, −1, −2. Notably, in both relay types astigmatism and medial focus surface have the same behavior such that for m < 0, astigmatism is always non-zero and minimal at ɛ = 0 (i.e. 2f₁ + 2f₂ configuration) and medial focus surface can be compensated when $ε = \pm f_{1} / \sqrt{- m}$ . Otherwise, there is a significant difference of the behavior of aberrations between both conjugates. In the type 2 relay, spherical aberration is unaffected by a change in d₁ while coma is linearly dependent with ɛ and thus cancels out for the 2f₁ + 2f₂ configuration. Whereas in the type 1 relay, coma can have up to three roots that are functions of magnification and the focal length of the first mirror, and spherical aberration has four roots when m > 0 and no roots when m < 0 except for when m = −1 where it has two roots. This suggests that the performance of the type 1 relay will have greater benefit from moving away from ɛ = 0.

Table 7. Seidel Coefficients, Their Respective Roots, and NAT Field Shifts ( $\overset{⇀}{σ}$ ) for a Modified 2f₁ + 2f₂ Type 1 Relay

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Table 8. Seidel Coefficients, Their Respective Roots and NAT Field Shifts ( $\overset{⇀}{σ}$ ) for a Modified 2f₁ + 2f₂ Type 2 Relay

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Fig. 13 Normalized absolute value of Seidel coefficients as a function of distance between pupil/object and the first optical element (with focal length f) for a modified 2f₁ + 2f₂ reflective relay.

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Included in both tables below are expressions for $\overset{⇀}{σ}$ in both relay types, respectively, with only the type 2 relay expressions a function of ɛ. The location of the nodes will be independent of ɛ if both $\overset{⇀}{σ}$ and the respective Seidel coefficients are independent of ɛ.

To demonstrate the role of $ε$ in the aberrations we plot the RMS wavefront field maps for design 1 (orthogonal geometry), see Table 6, of type 1 and type 2 with an ɛ of −250, 0, and 400 mm evaluated for 500 nm light in Fig. 14. Clearances δ₁ and δ₂ were maintained between 20 and 30 mm by changing $θ_{y_{1}}$ to 3.8° for ɛ = −250 mm and $θ_{x_{2}}$ to 6.0° for ɛ = 400 mm, respectively. Changing ɛ did not improve the RMS wavefront for either relay type except in the type 1 relay when ɛ = 400 mm. For this design example varying d₁ can lead to improved RMS wavefront and should be considered as an added variable for optimization of field-averaged wavefront variance.

Fig. 14 RMS wavefront field maps of an example system of type 1 (top row) and type 2 (bottom row) at three different ɛ values: −250, 0, and 400 mm (columns 1-3).

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4.5 Sensitivity of the 2f₁ + 2f₂ type 2 relay to defocus

An important variation of 2f₁ + 2f₂ type 2 relay is one in which the object is not at exactly at infinity, i.e. marginal angle is not collimated. This could occur unintentionally due to system misalignment or intentionally such as in ophthalmic imaging systems in which the eye’s defocus leads to a change in the marginal ray angle, termed vergence. Let us now consider such a system with a fixed entrance pupil diameter and input marginal ray angle φ_v as depicted below in Fig. 15. As Table 9 shows, all the Seidel coefficients vary with the product of the marginal ray angle and the focal length of the first mirror in the relay. For small vergences (i.e., $f_{1} φ_{v}$ << 1), the linear terms on φ_v dominate, which means coma, astigmatism and medial focus are the most sensitive to small departures from collimation. The field shifts $\overset{⇀}{σ}$ do not change with vergence, however, the NAT coefficients do since the Seidel coefficients depend on vergence.

Fig. 15 Type 2 relay input marginal rays with deviation from collimation (i.e. vergence) represented by φ_v.

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Table 9. Seidel Coefficients, Roots, and Field Shift Vectors $\overset{⇀}{σ}$ for a Defocused 2f₁ + 2f₂ Type 2 Relay

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To illustrate the effect of vergence, the RMS wavefront field maps for design 2 type 2 (see Table 6) in both in-plane and orthogonal geometries are shown in Fig. 16 below for a change in marginal ray angle of ± 0.5°, ± 0.2°, and 0°, evaluated for 500 nm light. These angles were chosen to represent a vergence range of the eye equivalent to ± 2 D, for an 8 mm pupil, which covers a small portion of the myopic and hyperopic human population, but demonstrates the substantial change in RMS wavefront for ophthalmic systems as a result of vergence. Furthermore, the change in RMS wavefront is asymmetric, such that the field-averaged RMS wavefront for 2 D and −2 D are ~6 and ~2 times greater than at 0 D, respectively. This suggests potentially biasing the collimation towards a converging beam for reduced sensitivity of the performance under alignment error.

Fig. 16 RMS wavefront field maps evaluated for 500 nm light for a type 2 relay with the parameters of design 2 in-plane (row 1) and orthogonal (row 2) geometries for 2D, 1D, 0D, −1D, and −2D of angle error (columns 1-5).

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4.6 Multiple conjugate systems

An application of the formulae and method discussed above is the design of systems that require correction of multiple conjugates simultaneous, particularly in the image and pupil (type 1 and 2) [60]. The angles in such designs can be optimized by minimizing the metric

{{\bar{W_{var}}}_{_{\overset{⇀}{H}}}}_{T o t a l}^{N R S} = \sum_{i}^{} ω_{i} {{\bar{W_{var}}}_{_{\overset{⇀}{H}}}}_{i}^{N R S},

in which i is the conjugate index, and ω_i is a positive scalar denoting the desired weight assigned to each conjugate. For a system in which image and pupil performance are optimized Eqs. (14) and (15) can be used.

5. Conclusion

A method for finding starting geometries for two mirror unobscured relays using nodal aberration theory was demonstrated. The minimization of the field-averaged wavefront variance calculated using only NAT coefficients was proposed as an optimization method. When tested in sample reflective 2f₁ + 2f₂ relays, the optimal angles were in close agreement with those obtained with real ray tracing software. Comparison of the NAT coefficients for in-plane and orthogonal folding indicates that one geometry is preferable over the other according to the amplitude of constant astigmatism relative to the other third order NAT coefficients. Also, we illustrated how the object or pupil distance from the first mirror in a relay telescope can be used to change the magnitude of the Seidel and NAT coefficients, and thus can be used to reduce the total field-averaged wavefront variance. Additionally, it was shown that the Seidel coefficients of the infinite conjugate reflective 2f₁ + 2f₂ relay vary quadratically with collimation error to a first approximation. Finally, the optimization of multiple conjugate systems through a weighted combination of wavefront variances is suggested.

Appendix A modified 2f₁ + 2f₂ relay

Here, we define the modified 2f₁ + 2f₂ relay as formed by two infinitely thin optical elements such that image telecentricity is achieved for any type 1 relay, or equivalently, marginal ray collimation is achieved for a type 2 relay. For such a system, the paraxial magnification is,

m = \frac{h_{f}}{h_{0}} = \frac{f_{1} f_{2}}{d_{1} (d_{2} - f_{1} - f_{2}) + f_{1} (f_{2} - d_{2})},

where d₁ is the distance between the aperture and first element for a type 2 relay or the object and first element in a type 1 relay, and d₂ is the separation between the elements. We now impose that the exit chief ray or marginal ray of the type 1 and type 2 relays, respectively, is parallel to the optical axis, which is,

u_{a_{f}} = \frac{d_{2} - f_{1} - f_{2}}{f_{1} f_{2}} h_{0} = 0

This condition can only be met if d₂ = f₁ + f₂, which simplifies Eq. (17) to $m = - f_{2} / f_{1}$ for both type 1 and type 2 relays. Given the current constraints, the exit angle of the exit marginal ray or chief ray for the type 1 and type 2 relays, respectively, is $u_{b_{f}} = m α_{0}$ , where α₀ is the initial ray and is assumed to be positive. If we also impose that the conjugate plane is real, this requires $u_{b_{f}} < 0$ and thus m < 0, which subsequently means that f₁, f₂ > 0.

Appendix B field-averaged wavefront variance extrema in 2f₁ + 2f₂ relays

The type 1 relay total field-averaged wavefront variance Eq. (14) has three roots,

θ_{y_{2}} = \frac{3 α_{0}^{2} f_{1}^{2} + 4 m h_{0}^{2}}{16 m θ_{y_{1}} f_{1}^{2}},

θ_{x_{2}} = \pm \frac{\sqrt{- {(3 α_{0}^{2} f_{1}^{2} + 4 m h_{0}^{2})}^{2} - 32 f_{1}^{2} θ_{y_{1}}^{2} (3 α_{0}^{2} f_{1}^{2} + 4 m^{2} h_{0}^{2}) - 256 m^{3} f_{1}^{4} θ_{y_{1}}^{4}}}{16 m θ_{y_{1}} f_{1}^{2}},

and

θ_{y_{2}} = \frac{- 48 {(2)}^{1 / 3} [- 4 m^{4} f_{1}^{2} h_{0}^{2} + f_{1}^{4} (- 3 α_{0}^{2} m^{2} + 8 m^{5} θ_{y_{1}}^{2})] + 16 {(3)}^{2 / 3} {(V)}^{2 / 3}}{96 {(2)}^{2 / 3} {(3)}^{1 / 3} m^{2} f_{1}^{2} {(V)}^{1 / 3}},

θ_{x_{2}} = 0,

where

V = - θ_{y_{1}} (54 m^{6} f_{1}^{6} α_{0}^{2} + 72 m^{7} f_{1}^{4} h_{0}^{2}) + \sqrt{6 m^{6} f_{1}^{6} {54 m^{2} f_{1}^{2} θ_{y_{1}}^{2} {[3 f_{1}^{2} α_{0}^{2} + 4 m h_{0}^{2}]}^{2} + {[4 m^{2} h_{0}^{2} + f_{1}^{2} (3 α_{0}^{2} - 8 m^{3} θ_{y_{1}}^{2})]}^{3}}} .

The corresponding solutions for the type 2 relay (Eq. (15)) are

θ_{y_{2}} = \frac{4 α_{0}^{2} f_{1}^{2} + 3 m h_{0}^{2}}{16 m θ_{y_{1}} f_{1}^{2}},

θ_{x_{2}} = \pm \frac{\sqrt{- 9 m^{2} h_{0}^{4} - 24 m f_{1}^{2} h_{0}^{2} (α_{0}^{2} + 4 m θ_{y_{1}}^{2}) - 16 f_{1}^{4} (α_{0}^{4} + 8 α_{0}^{2} θ_{y_{1}}^{2} + 16 m θ_{y_{1}}^{4})}}{16 m θ_{y_{1}} f_{1}^{2}},

while the third solution is

θ_{y_{2}} = \frac{- 48 {(2)}^{1 / 3} m^{2} f_{1}^{2} [3 m^{2} h_{0}^{2} + 4 f_{1}^{2} (α_{0}^{2} - 2 m θ_{y_{1}}^{2})] + 16 {(3)}^{2 / 3} {(V)}^{2 / 3}}{96 {(2)}^{2 / 3} {(3)}^{1 / 3} m^{2} f_{1}^{2} {(V)}^{1 / 3}},

θ_{x_{2}} = 0,

where,

V = - θ_{y_{1}} (72 m^{4} f_{1}^{6} α_{0}^{2} + 54 m^{5} f_{1}^{4} h_{0}^{2}) + \sqrt{6 m^{6} f_{1}^{6} {54 m^{2} f_{1}^{2} θ_{y_{1}}^{2} {[4 f_{1}^{2} α_{0}^{2} + 3 m h_{0}^{2}]}^{2} + {[3 m^{2} h_{0}^{2} + 4 f_{1}^{2} (α_{0}^{2} - 2 m θ_{y_{1}}^{2})]}^{3}}} .

Appendix C roots of $W_{131}$ for a modified 2f₁ + 2f₂ type 1 relay

The expression for W₁₃₁ of a type 1 modified 2f₁ + 2f₂ relay results in three roots. Two of the three roots of W₁₃₁ are imaginary, and thus only the non-imaginary root is shown,

ε = \frac{- 2 {(3)}^{1 / 3} m^{3} f_{1}^{2} + 2^{1 / 3} m^{8 / 3} f_{1}^{2} {[\sqrt{3 (27 m^{2} + 58 m + 27)} - 9 (m + 1)]}^{2 / 3}}{6^{2 / 3} m^{10 / 3} f_{1} {[\sqrt{3 (27 m^{2} + 58 m + 27)} - 9 (m + 1)]}^{1 / 3}} .

Funding

National Eye Institute (EY025231, EY025477, EY026877); Research to Prevent Blinding (Departmental Award).

Acknowledgments

The authors would like to thank Zemax for providing a student license to OpticStudio.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Aberration	Wavefront contribution	Coefficients	Field Dependence
Spherical	$W_{040} {(\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})}^{2}$	$W_{040} = \sum_{j} W_{040_{j}}$	Constant
Coma	$[(W_{131} \overset{⇀}{H} - {\overset{⇀}{A}}_{131}) \cdot \overset{⇀}{ρ}] (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ})$	$W_{131} = \sum_{j} W_{131_{j}}$ ${\overset{⇀}{A}}_{131} = \sum_{j} W_{131_{j}} {\overset{⇀}{σ}}_{j}$
Astigmatism	$\frac{1}{2} [W_{222} {\overset{⇀}{H}}^{2} - 2 \overset{⇀}{H} {\overset{⇀}{A}}_{222} + {\overset{⇀}{B}}_{222}^{2}] \cdot {\overset{⇀}{ρ}}^{2}$	$W_{222} = \sum_{j} W_{222_{j}}$ ${\overset{⇀}{A}}_{222} = \sum_{j} W_{222_{j}} {\overset{⇀}{σ}}_{j}$ ${\overset{⇀}{B}}_{222}^{2} = \sum_{j} W_{222_{j}} {\overset{⇀}{σ}}_{j}^{2}$
Medial Focus Surface	$\begin{array}{l} [W_{220_{M}} (\overset{⇀}{H} \cdot \overset{⇀}{H}) - 2 (\overset{⇀}{H} \cdot {\overset{⇀}{A}}_{220_{M}}) \\ + B_{220_{M}}] (\overset{⇀}{ρ} \cdot \overset{⇀}{ρ}) \end{array}$	$W_{220_{M}} = \sum_{j} W_{220_{M}_{j}}$ ${\overset{⇀}{A}}_{220_{M}} = \sum_{j} W_{220_{M}_{j}} {\overset{⇀}{σ}}_{j}$ $B_{220_{M}} = \sum_{j} W_{220_{M}_{j}} {\overset{⇀}{σ}}_{j} \cdot {\overset{⇀}{σ}}_{j}$

Parameter	System 1	System 2
m	−1.0	−2.0
$θ_{y_{1}}$	2.0°	2.0°
$θ_{y_{2}}$	0.0°	−1.2°
$θ_{x_{2}}$	2.0°	0.0°

	Type 1 (Finite conj.)	Type 2 (Infinite conj.)
$W_{040}$	$\frac{f_{1} α_{0}^{4}}{32} (\frac{m^{3} - 1}{m^{3}})$	$\frac{h_{0}^{4}}{32 f_{1}^{3}} (1 - m)$
$W_{131}$	$\frac{h_{0} α_{0}^{3}}{8} (\frac{1 - m^{2}}{m^{2}})$	0
$W_{222}$	$\frac{h_{0}^{2} α_{0}^{2}}{8 f_{1}} (\frac{m - 1}{m})$	$\frac{α_{0}^{2} h_{0}^{2}}{8 f_{1}} (\frac{m - 1}{m})$
$W_{220_{M}}$	$- \frac{h_{0}^{2} α_{0}^{2}}{8 f_{1}} (\frac{m - 1}{m})$	$- \frac{α_{0}^{2} h_{0}^{2}}{8 f_{1}} (\frac{m - 1}{m})$

	Type 1 (Finite conjugate)	Type 2 (Infinite conjugate)
${\overset{⇀}{σ}}_{1}, {\overset{⇀}{σ}}_{2}$	$\frac{2 f_{1}}{h_{0}} [\begin{matrix} 0 & - θ_{y_{1}} & 0 \end{matrix}], \frac{2 f_{1}}{h_{0}} [\begin{matrix} θ_{x_{2}} & θ_{y_{2}} & 0 \end{matrix}]$	$\frac{2}{α_{0}} [\begin{matrix} 0 & θ_{y_{1}} & 0 \end{matrix}], \frac{2 m}{α_{0}} [- \begin{matrix} θ_{x_{2}} & - θ_{y_{2}} & 0 \end{matrix}]$
$A_{131}^{2}$	$\frac{f_{1}^{2} α_{0}^{6}}{16} [\frac{{(m^{2} θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}}{m^{4}}]$	$\frac{h_{0}^{6}}{16 f_{1}^{4}} [{(θ_{y_{1}} + m θ_{y_{2}})}^{2} + m^{2} θ_{x_{2}}^{2}]$
$A_{222}^{2} = A_{220_{M}}^{2}$	$\frac{h_{0}^{2} α_{0}^{4}}{16} [\frac{{(m θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}}{m^{2}}]$	$\frac{α_{0}^{2} h_{0}^{4}}{16 f_{1}^{2}} [{(θ_{y_{1}} + θ_{y_{2}})}^{2} + θ_{x_{2}}^{2}]$
$B_{222}^{4}$	$\frac{f_{1}^{2} α_{0}^{4}}{4} [\frac{{(m θ_{y_{1}}^{2} + θ_{x_{2}}^{2} - θ_{y_{2}}^{2})}^{2} + 4 θ_{x_{2}}^{2} θ_{y_{2}}^{2}}{m^{2}}]$	$\frac{h_{0}^{4}}{4 f_{1}^{2}} [{(θ_{y_{1}}^{2} + m θ_{x_{2}}^{2} - m θ_{y_{2}}^{2})}^{2} + 4 m^{2} θ_{x_{2}}^{2} θ_{y_{2}}^{2}]$

Parameter	Design 1	Design 2	Design 3
f₁ (mm)	500	250	100
m	−0.5	−2.0	−2.0
h₀ (mm)	4	8	10
$α_{0}$ (°)	1.0	0.5	1.0
$θ_{y_{1}}$ (°)	3.0	3.0	3.0
${(θ_{x_{2}}^{N A T}, θ_{y_{2}}^{N A T})}_{T 1}$	( ± 1.70, −0.11)	( ± 4.03, 0.27)	(0.00, 2.41)
${(θ_{x_{2}}^{N A T}, θ_{y_{2}}^{N A T})}_{T 2}$	( ± 3.99, −0.15)	( ± 1.78, 0.20)	(0.00, 0.74)

Design of two spherical mirror unobscured relay telescopes using nodal aberration theory

Abstract

1. Introduction

2. Background

2.1 Modified 4f relay

2.2 Unobscured two-mirror relay

3. Third order aberration theory

3.1 Rotationally symmetric optical systems

3.2 Non-rotationally symmetric optical systems

3.3 Field shift calculation for tilted spherical mirrors

3.4 Wavefront variance as a performance metric

4. Two spherical mirror unobscured 2f₁ + 2f₂ relay

4.1 The 2f₁ + 2f₂ relay

4.2 Example designs

4.3 In-plane and orthogonal geometries

4.4 Modified 2f₁ + 2f₂ relay

4.5 Sensitivity of the 2f₁ + 2f₂ type 2 relay to defocus

4.6 Multiple conjugate systems

5. Conclusion

Appendix A modified 2f₁ + 2f₂ relay

Appendix B field-averaged wavefront variance extrema in 2f₁ + 2f₂ relays

Appendix C roots of $W_{131}$ for a modified 2f₁ + 2f₂ type 1 relay

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (16)

Tables (9)

Equations (29)

Optics Express

	Folding	m	$θ_{y_{1}}$	$θ_{y_{2}}$	$θ_{x_{2}}$
Design 1	In-plane	−1.0	2.0	−2.0	0.0
Design 1	Orthogonal	−1.0	2.0	0.0	2.0
Design 2	In-plane	−0.5	2.0	−3.5	0.0
Design 2	Orthogonal	−0.5	2.0	0.0	3.5

	Type 1 (Finite conjugate)	Roots
$W_{040}$	$\frac{α_{0}^{4}}{32 m^{3} f_{1}^{3}} (1 - m) [m^{3} ε^{4} + 2 m^{2} ε^{2} f_{1}^{2} - (1 + m + m^{2}) f_{1}^{4}]$	$ε = \pm f_{1} \sqrt{- \frac{1}{m} \pm \frac{(1 + m)}{\sqrt{m^{3}}}}$
$W_{131}$	$\frac{h_{0} α_{0}^{3}}{8 m^{2} f_{1}^{3}} (1 - m) [m^{2} ε^{3} + m ε f_{1}^{2} + (1 + m) f_{1}^{3}]$	(see Appendix C)
$W_{222}$	$\frac{h_{0}^{2} α_{0}^{2}}{8 m f_{1}^{3}} (1 - m) (m ε^{2} - f_{1}^{2})$	$ε = \pm f_{1} / \sqrt{m}$
$W_{220_{M}}$	$\frac{h_{0}^{2} α_{0}^{2}}{8 m f_{1}^{3}} (1 - m) (m ε^{2} + f_{1}^{2})$	$ε = \pm f_{1} / \sqrt{- m}$
${\overset{⇀}{σ}}_{1}, {\overset{⇀}{σ}}_{2}$	$\frac{2 f_{1}}{h_{0}} [\begin{matrix} 0 & - θ_{y_{1}} & 0 \end{matrix}], \frac{2 f_{1}}{h_{0}} [\begin{matrix} θ_{x_{2}} & θ_{y_{2}} & 0 \end{matrix}]$	N/A

	Type 2 (Infinite conjugate)	Roots
$W_{040}$	$\frac{h_{0}^{4}}{32 f_{1}^{3}} (1 - m)$	none
$W_{131}$	$\frac{α_{0} h_{0}^{3}}{8 f_{1}^{3}} (1 - m) ε$	$ε = 0$
$W_{222}$	$\frac{α_{0}^{2} h_{0}^{2}}{8 m f_{1}^{3}} (1 - m) (m ε^{2} - f_{1}^{2})$	$ε = \pm f_{1} / \sqrt{m}$
$W_{220_{M}}$	$\frac{α_{0}^{2} h_{0}^{2}}{8 m f_{1}^{3}} (1 - m) (m ε^{2} + f_{1}^{2})$	$ε = \pm f_{1} / \sqrt{- m}$
${\overset{⇀}{σ}}_{1}, {\overset{⇀}{σ}}_{2}$	$\frac{2 f_{1}}{α_{0} (f_{1} - ε)} [\begin{matrix} 0 & θ_{y_{1}} & 0 \end{matrix}], \frac{2 m f_{1}}{α_{0} (f_{1} - m ε)} [\begin{matrix} - θ_{x_{2}} & - θ_{y_{2}} & 0 \end{matrix}]$	N/A

	Type 2	Roots
$W_{040}$	$\frac{h_{0}^{4}}{32 f_{1}^{3}} [\frac{m^{3} (1 - m) + 2 m^{2} f_{1}^{2} φ_{v}^{2} (1 - m) + f_{1}^{4} φ_{v}^{4} (m^{3} - 1)}{m^{3}}]$	$\pm \frac{1}{f_{1}} \sqrt{\frac{m^{2} \pm (m + 1) \sqrt{m^{3}}}{m^{2} + m + 1}}$
$W_{131}$	$\frac{h_{0}^{3} α_{0}}{8 f_{1}} {\frac{m^{2} (m - 1) + f_{1} φ_{v} [m^{3} - m - f_{1} φ_{v} (m^{3} - 1)]}{m^{3}}} φ_{v}$	$0, \frac{m [1 + m \pm \sqrt{5 + m (6 + 5 m)}]}{2 f_{1} (m^{2} + m + 1)}$
$W_{222}$	$\frac{α_{0}^{2} h_{0}^{2}}{8 f_{1}^{3}} {\frac{m^{2} (m - 1) + f_{1} φ_{v} [2 m - 2 m^{3} + f_{1} φ_{v} (m^{3} - 1)]}{m^{3}}}$	$\frac{m (m + 1) \pm \sqrt{m^{3}}}{f_{1} (m^{2} + m + 1)}$
$W_{220_{M}}$	$\frac{α_{0}^{2} h_{0}^{2}}{8 f_{1}^{3}} {\frac{m^{2} (1 - m) + f_{1} φ_{v} [2 m - 2 m^{3} + f_{1} φ_{v} (m^{3} - 1)]}{m^{3}}}$	$\frac{m (m + 1) \pm m \sqrt{2 + m (3 + 2 m)}}{f_{1} (m^{2} + m + 1)}$

Abstract

1. Introduction

2. Background

2.1 Modified 4f relay

2.2 Unobscured two-mirror relay

3. Third order aberration theory

3.1 Rotationally symmetric optical systems

3.2 Non-rotationally symmetric optical systems

3.3 Field shift calculation for tilted spherical mirrors

3.4 Wavefront variance as a performance metric

4. Two spherical mirror unobscured 2f1 + 2f2 relay

4.1 The 2f1 + 2f2 relay

4.2 Example designs

4.3 In-plane and orthogonal geometries

4.4 Modified 2f1 + 2f2 relay

4.5 Sensitivity of the 2f1 + 2f2 type 2 relay to defocus

4.6 Multiple conjugate systems

5. Conclusion

Appendix A modified 2f1 + 2f2 relay

Appendix B field-averaged wavefront variance extrema in 2f1 + 2f2 relays

Appendix C roots of W131 for a modified 2f1 + 2f2 type 1 relay

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (16)

Tables (9)

Equations (29)

Optics Express

4. Two spherical mirror unobscured 2f₁ + 2f₂ relay

4.1 The 2f₁ + 2f₂ relay

4.4 Modified 2f₁ + 2f₂ relay

4.5 Sensitivity of the 2f₁ + 2f₂ type 2 relay to defocus

Appendix A modified 2f₁ + 2f₂ relay

Appendix B field-averaged wavefront variance extrema in 2f₁ + 2f₂ relays

Appendix C roots of $W_{131}$ for a modified 2f₁ + 2f₂ type 1 relay