Algorithm for calculating anastigmatic three-mirror telescopes

Abstract

A simple algorithm for calculating three-mirror telescopes with all conic surfaces free of spherical aberration, coma, astigmatism, and field curvature is proposed. The algorithm is based on the theory of aberrations of the third order. The initial parameters are the effective focal length, mirror diameters, and system length. The result is a design with a flat field of view with a diameter of more than 1° at sub-arcsecond image quality, which is close to the theoretically best. If necessary, the latter is achieved through slight optimization.

Appendix A: The Seidel-Schwarzschild theory of third-order aberrations, as applied to three-mirror telescopes

A detailed description of the relevant analytical calculations seems appropriate both to justify the proposed algorithm and in view of possible applications of this apparatus for similar problems. The initial relations of geometric optics used below are described in a number of manuals, in particular, in the monographs of Welford [13], Geary [14], and Churilovsky [15]. Our presentation is closest to the last book.

1.1 A.1 First-order equations for the marginal ray

Suppose, as is usually the case in practice, that the entrance pupil is located at the primary mirror. For completeness, recall the basic notation: D, D₂, and D₃ are diameters of the primary, the secondary, and the tertiary mirrors, respectively; the normalizing parameter H ≡ D/2; H_k, T_k, and R_k are, correspondigly, the heights of the marginal ray on mirror surfaces (H₁ = H), the distances between the mirrors, and the paraxial radiuses of curvature. It is convenient to enter the following dimensionless quantities:

$$ h_{k} \equiv H_{k}/H, \quad t_{k} \equiv T_{k}/H, \quad r_{k} \equiv R_{k}/H; \qquad k = 1,2,3. $$

(A1)

The most attractive version of the three-mirror system is considered here, for which

$$ t_{1} < 0, \quad t_{2} > 0, \quad t_{3} < 0 $$

(A2)

(see Fig. 1). The set of [α₁ = 0,α₂,α₃,α₄] represents the tangents of the angles indicated in Fig. 1; by definition of the effective focal length F, we have

$$ \alpha_{4} = -H/F = -1/f, $$

(A3)

where f ≡ F/H.

The equations for the marginal ray that passes through a sequence of media with refractive indices {n_k} are given by:

$$ \left\{\begin{array}{l} h_{k+1} = h_{k} -\alpha_{k+1} t_{k}, \\ \alpha_{k+1} = \nu_{k+1} \alpha_{k} / \nu_{k} + (\nu_{k}-\nu_{k+1}) h_{k}/(\nu_{k} r_{k}), \end{array}\right. $$

(A4)

where ν_k ≡ 1/n_k. In our case,

$$ \nu_{1} = \nu_{3} = 1, \qquad \nu_{2} = \nu_{4} = -1, $$

(A5)

so the equations for the heights of the rays on the mirrors surfaces are:

$$ \left\{\begin{array}{l} h_{1} = 1, \\ h_{2} = 1 - \alpha_{2} t_{1}, \\ h_{3} = h_{2} - \alpha_{3} t_{2}, \\ h_{4} = h_{3} - \alpha_{4} t_{3}, \end{array} \right. $$

(A6)

whereas the (A4) for the angles take the form:

$$ \left\{\begin{array}{l} \alpha_{1} = 0, \\ \alpha_{2} = 2/r_{1}, \\ \alpha_{3} = -\alpha_{2} + 2 h_{2}/r_{2}, \\ \alpha_{4} = -\alpha_{3} + 2 h_{3}/r_{3}. \end{array} \right. $$

(A7)

Since h₄ = 0, the last of (A6) and (A3) give for the back focal length:

$$ t_{3} = h_{3}/\alpha_{4} = -f h_{3}. $$

(A8)

It follows from (A8) and (A2), that f and h₃ must have the same sign, i.e., h₃ > 0 for F > 0, and h₃ < 0 for F < 0. Therefore, in the latter case, the marginal ray intersects the optical axis with the formation of an intermediate image. As far as the diameters of all mirrors are assumed to be positive, we can write for arbitrary F:

$$ h_{2} = D_{2}/D, \qquad h_{3} = \text{sign}(F)\cdot D_{3}/D, $$

(A9)

which coincides with (3) of the main text.

1.2 A.2 The Petzval condition

A necessary condition for the focal surface of a system with N optical media to be flat is the Petzval condition:

$$ \sum\limits_{k=1}^{N} \frac{\nu_{k+1}-\nu_{k}}{r_{k}} = 0, $$

(A10)

where ν_k ≡ 1/n_k (Born and Wolf [16], Section 5.5.3). The insufficiency of the Petzval condition means that astigmatism must be corrected simultaneously to reach the flat focal surface. We will require below that all third-order aberrations, including astigmatism and Petzval curvature, be equal to zero, but now it is convenient to consider the condition (A10) first. Given (A5), we obtain for a three-mirror system:

$$ -1/r_{1} + 1/r_{2} - 1/r_{3} = 0. $$

(A11)

This means that the sum of the optical powers of all the mirror components of the system is zero.

Further calculations are simplified when passing to variables {α_k,h_k,t_k}. Taking into account (A7), we write down the Petzval condition as follows:

$$ -\alpha_{2} +\frac{\alpha_{2}+\alpha_{3}}{h_{2}} - \frac{\alpha_{3}+\alpha_{4}}{h_{3}} = 0. $$

(A12)

The other two useful forms of the Petzval condition, namely,

$$ \alpha_{2} = \frac{t_{2}{{\alpha}_{3}^{2}} + h_{2}\alpha_{4}}{(1-h_{2})h_{3}} , $$

(A13)

and

$$ t_{2}{{\alpha}_{3}^{2}} + t_{1}t_{2}{{\alpha}_{2}^{2}}\alpha_{3} + h_{2}(\alpha_{4}-t_{1}{{\alpha}_{2}^{2}}) = 0, $$

(A14)

can be obtained using the third equation of the system (A6).

1.3 A.3 The first part of the algorithm

In total, there are 13 parameters that describe a three-mirror system with accuracy to a scale factor:

$$ \left\{\begin{array}{lll} h_{2}, & h_{3}, & \\ t_{1}, & t_{2}, & t_{3}, \\ \alpha_{2}, & \alpha_{3}, & \\ r_{1}, & r_{2}, & r_{3}, \\ \sigma_{1}, & \sigma_{2}, & \sigma_{3}. \end{array} \right. $$

(A15)

To find them, we have 6 equations (A6)–(A7) of the first-order ray optics and the Petzval condition of the third-order theory of aberrations, i.e., 7 relations. Three more equations for finding the conic constants of mirrors will give the requirement to correct third-order aberrations, which are discussed in the next section. Thus, we have the right to set any three parameters of our choice.

It seems that the simple and reliable way is to set the distance between the secondary and tertiary mirrors together with their diameters, that is, t₂ = T₂/H and the parameters h₂, h₃ by means of (A9). Then (4) for α₃ of the main text follows from (A6), whereas the Petzval condition in the form of (A13) allows one to find α₂. Finally, the expressions for t₁,t₃ and all the radii of curvature follow from (A6) and (A7).

Lee and Yu [7] chose a different way when considering first-order equations. Namely, they consider (in our notation) t₁,t₂ and α₂ (or r₁) known, and then found α₃ by solving the quadratic equation (A14). Checking the acceptability of the initial parameters is given by the condition of non-negativeness of the determinant of this equation. Since the quadratic equation has two roots, a physically acceptable solution should be chosen at the end. From our point of view, the angle of inclination of the marginal ray between the secondary and tertiary mirrors, i.e. α₃, should be determined more rigidly, because its small variations lead to significant changes in the size of the tertiary mirror and in the entire optical layout.

Returning to our procedure, it should be noted that, for a given diameter of the secondary mirror, the choice of the diameter of the tertiary mirror is not completely free. We have in mind (1), the condition of the negativity of all radii of curvature. It is easy to verify that such restrictions arise only for systems with a positive effective focal length F. In this case, the requirements R₁ < 0 and R₃ < 0 are more stringent than R₂ < 0. The first of these conditions leads to the inequality $h_{3} < h_{2}+\sqrt {\tau h_{2}}$, where τ ≡ t₂/(2ϕ) = T₂/F. Next condition leads to inequality α₃ + α₄ < 0, or h₂ − h₃ < τ. Considering the non-negativity of h₃ for F > 0, we come to (11) and (12) of the main text.

1.4 A.4 The conic constants

The theory of aberrations used here proceeds from eikonal in the form that Karl Schwarzschild [1] gave to this fundamental function (Born and Wolf [16], §5.2; Welford [13], Chapter 7; Churilovsky [15], §29). The last of the named authors gives general expressions for third-order coefficients of spherical aberration (B), coma (K), astigmatism (C), and field curvature (C + D) for an optical system with conic surfaces at an arbitrary position of the entrance pupil. In the particular case of a three-mirror telescope with the entrance pupil located at the primary mirror (which is indicated below by the index ‘0’), these expressions take the form:

$$ \left\{\begin{array}{ll} B_{0} & = \frac{1}{2} {\sum}_{k=1}^{3} h_{k}Q_{k}, \\ & \\ K_{0} & = \frac{1}{2} {\sum}_{k=2}^{3} h_{k} S_{k} Q_{k} - \frac{1}{2} {\sum}_{k=1}^{3} W_{k}, \\ & \\ C_{0} & = \frac{1}{2} {\sum}_{k=2}^{3} h_{k} {S_{k}^{2}} Q_{k} - {\sum}_{k=2}^{3} S_{k} W_{k} + L/2, \\ & \\ D_{0} & = C_{0} - L/2, \end{array} \right. $$

(A16)

where

$$ \left\{\begin{array}{ll} Q_{1} & = T_{1} \sigma_{1} + P_{1}, \\ Q_{2} & = T_{2} \sigma_{2} + P_{2}, \\ Q_{3} & = T_{3} \sigma_{3} + P_{3}; \end{array} \right. $$

(A17)

$$ \left\{\begin{array}{ll} T_{1} & = - {\alpha_{2}^{3}}/4, \\ T_{2} & = (\alpha_{2} + \alpha_{3})^{3}/4, \\ T_{3} & = -(\alpha_{3} + \alpha_{4})^{3}/4; \end{array} \right. $$

(A18)

$$ \left\{\begin{array}{ll} P_{1} & = T_{1}, \\ P_{2} & = (\alpha_{2} + \alpha_{3})(\alpha_{2} - \alpha_{3})^{2}/4, \\ P_{3} & = -(\alpha_{3} + \alpha_{4})(\alpha_{3} - \alpha_{4})^{2}/4; \end{array} \right. $$

(A19)

$$ \left\{\begin{array}{ll} S_{2} & = -t_{1}/h_{2}, \\ S_{3} & = t_{2}/(h_{2}h_{3}) -t_{1}/h_{2}; \end{array} \right. $$

(A20)

$$ \left\{\begin{array}{ll} W_{1} & = {{\alpha}_{2}^{2}}/2, \\ W_{2} & = ({{\alpha}_{3}^{2}} - {{\alpha}_{2}^{2}})/2, \\ W_{3} & = ({{\alpha}_{4}^{2}} - {{\alpha}_{3}^{2}})/2; \end{array} \right. $$

(A21)

$$ L = -\alpha_{2} +\frac{\alpha_{2}+\alpha_{3}}{h_{2}} - \frac{\alpha_{3}+\alpha_{4}}{h_{3}} . $$

(A22)

As can be seen, the expression (A22) for the coefficient L coincides with the left side of the Petzval condition (A12), which we have already equated to zero. Thus, equating the remaining aberration coefficients B₀, K₀, and C₀ in (A16) to zero leads to the simultaneous linear equations for Q₁, Q₂, and Q₃:

$$ \left\{\begin{array}{ll} & {\sum}_{k=1}^{3} h_{k}Q_{k} = 0, \\ & \\ & {\sum}_{k=2}^{3} h_{k} S_{k} Q_{k} = {{\alpha}_{4}^{2}}/2, \\ & \\ & {\sum}_{k=2}^{3} h_{k} {{S}_{k}^{2}} Q_{k} = 2{\sum}_{k=2}^{3} S_{k} W_{k}, \end{array} \right. $$

(A23)

the solution of which is:

$$ \left\{\begin{array}{ll} Q_{1} & = -h_{2}Q_{2} -h_{3}Q_{3}, \\ Q_{2} & = \left[ S_{3}{\alpha_{4}^{2}}/2-2(S_{2}W_{2}+S_{3}W_{3})\right]/ \left[ h_{2}S_{2}(S_{3}-S_{2})\right], \\ Q_{3} & = \left[ S_{2}{\alpha_{4}^{2}}/2-2(S_{2}W_{2}+S_{3}W_{3})\right]/ \left[ h_{3}S_{3}(S_{2}-S_{3})\right]. \end{array} \right. $$

(A24)

The required representation of the conic constants of the mirrors as functions of already known parameters follows from (A17) and the first of equations (A19):

$$ \sigma_{1} = -1+\frac{Q_{1}}{T_{1}} , \qquad \sigma_{2} = \frac{Q_{2}-P_{2}}{T_{2}} , \qquad \sigma_{3} = \frac{Q_{3}-P_{3}}{T_{3}} . $$

(A25)

The final (8)–(10) of the main text can be obtained from the last formulas by successive substitutions and simplifications using (A18)–(A21).