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

Nodal aberration theory is used to calculate the third-order aberrations that result in image blur for an unobscured modified 4f relay (2f1 + 2f2) formed by two tilted spherical mirrors for objects at infinity (infinite conjugate) and near the front focal plane of the first mirror (finite conjugate). The field-averaged wavefront variance containing only non-rotationally symmetric aberration coefficients is then proposed as an optimization metric. Analytical and ray tracing optimization are demonstrated through sample designs. The particular cases of in-plane and orthogonal folding of the optical axis ray are discussed, followed by an analysis of a modified 2f1 + 2f2 relay in which the distance of the first mirror to the object or pupil is allowed to vary for aberration correction. The sensitivity of the infinite conjugate 2f1 + 2f2 relay to the input marginal ray angle is also examined. Finally, the optimization of multiple conjugate systems through a weighted combination of wavefront variances is proposed.


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][2][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][5][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][8][9][10][11][12][13][14][15][16][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][19][20][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 nonsymmetric optical systems. Early mathematical theories of non-symmetrical systems centered on small perturbations of rotationally symmetric systems [22][23][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][29][30]. This last approach has more recently been generalized and demonstrated in a variety of unobscured mirror systems [31][32][33][34][35][36][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][43][44][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 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.

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). 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, 21 m f f  , 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).
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) 0 h , input angle 0 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.

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 ( 1 0 y   ), and the second mirror around both the x-and y-axis ( 2 y  , 2 x  ). 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 xaxis, 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.
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

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], 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 Wklm 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 with the dot product defined as [50]).

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 j  so that H in Eq. (4) is replaced with j H   , and thus the wavefront aberration is given by where the coefficients Wklm 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 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 ( 131 A ) across the field. Astigmatism and the medial focus surface have constant shifts in the field, respectively 2 222 B and 220 , 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 ( 222 A ). Additionally, the medial focus surface has a longitudinal (axial) shift, given by 220 M B , that is of the same form as defocus. Third-order distortion is not considered here because it does not result in image blur.

Field shift calculation for tilted spherical mirrors
The field shift j  is a normalized vector contained in the image plane and given by [50], with j N being a unit vector normal to the object plane (pointing away from the direction of propagation), j R is a unit vector along the OAR incident on the surface (also pointing away from the direction of propagation), and j S 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 ij 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 j  for calculating the wavefront aberrations using Eq. (3), it must be converted to a 2-dimensional complex vector.

Wavefront variance as a performance metric
The wavefront variance Wvar at a field point H over a (circular) pupil is defined as [50], 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 [ (9) 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, where Wi is the weight of the operand, Ti is the target value, and Vi 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 which when substituted in Eq. (9) If one notes that the 220 M B term (see Table 1 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 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.

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  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  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.  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.

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. These are particular cases of the modified 2f1 + 2f2 relay coefficients in Table 7 and Table 8.
The NAT coefficients, calculated using the field shift vectors are shown below in Table 4 B 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 1  and 2  are proportional to 10 2 fh and 0 2  for the type 1 and 2 relays, respectively.
The NAT coefficients can be used to calculate the third-order expression for the NRS fieldaveraged wavefront variance using Eq. (13) for the type 2 relay. The local extrema of these metrics for a given 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.

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  x  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°). that minimize the NRS field-averaged wavefront variance for each conjugate. All angles in degrees. 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 μm 2 for type 1 designs 1-3, respectively, and 2x10 5 , 0.02, and 0.5 μm 2 for type 2 designs 1-3, respectively.
Next, we impose the mechanical considerations for mirror two as expressed by Eq. (2). The minimal clearance to avoid vignetting requires that 2 0   , however, to allow for mounting we set the clearance 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.

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 2 0 x   [12,13,[52][53][54][55][56][57][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 2 0 y   [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. Table 4 [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 12 , 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, 2 222 B , 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.

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 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.
Included in both tables below are expressions for  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  and the respective Seidel coefficients are independent of ɛ.
To demonstrate the role of  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 1 y  to 3.8° for ɛ = 250 mm and 2 x  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.  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.

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 φv 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., 1 v f  <<


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).

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  (16) in which i is the conjugate index, and ωi 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.

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 inplane 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.