## 3 Reflective Triplet (RT)  
Next, we describe how to convert the Cooke triplet into an off-axis, decentered, all-reflective system step by step. First, let us work on the first two elements. The corresponding reflective afocal optical system with the power of 0.5 is an afocal Mersenne design [16], illustrated in Figure 3a. Note that in this classical rotationally symmetrical system, the secondary mirror blocks some of the incoming beam bundle, introducing a so-called central obscuration. The two designs with the power of 0.5, that is, the two afocal systems formed by the first two elements of the Cooke triplet and the afocal Mersenne design, are identical when presented in the framework of the Y-\(\bar{\Upsilon}\) diagram, shown in Figure 2.  
The letter P for the primary denotes a positive primary element, while the letter S for the secondary denotes a negative one. In other words, element one in both the afocal portion of the Cooke triplet and the afocal Mersenne is positive, and the second one is negative. To mitigate the transmission loss caused by the central obscuration, the lower half of the afocal Mersenne is removed, as seen in Figure 3b, where the dashed line denotes the optical axis. This represents an off-axis optical system. The Y-\(\bar{\Upsilon}\) diagram of the afocal Mersenne design, with the optical stop located at the second mirror, is presented in Figure 4.  
Figure 3: Afocal Mersenne designs: (**a**) a classical rotationally symmetric system, where the secondary mirror blocks some of the incoming beam bundle, upon introducing a central obscuration; (**b**) an off-axis system is achieved upon slicing off the bottom half of the primary mirror, and recentering the secondary mirror. P and S stand for the primary and the secondary; and AS denotes the aperture stop, in this case at the secondary, that is, the negative mirror. OA denotes the optical axis. The description follows the one that accompanies Figure 2, except that the final mirror of Figure 2, which projects the image on the focal plane, is missing in Figure 4.  
If we rotate and/or decenter the second element slightly to avoid the beam obscuration and then add another positive mirror to form an image in the focal plane, we have now constructed a new conceptual optical design, that of an off-axis RT. Its Y-\(\bar{\text{Y}}\) diagram is the same as that of the classical Cooke triplet, presented in Figure 2.  
When the first two elements of the RT do not form an afocal beam after leaving the second element, we obtain the layout shown in Figure 5, disclosed as the US Patent 4,240,707 [17]. The key features of the design are that all elements share a common optical axis (Item 34). Both the primary and the secondary mirror may be cut out from the same optical blank, sometimes referred to as the parent element. Therefore, RT is top-bottom optically symmetrical with respect to the optical axis. The Cooke triplet, on the other hand, is left-right symmetrical about the central negative refractor. Next in Figure 5, the incoming beam, incident on the first mirror (Item 31), reflects off all three optical elements, without experiencing any transmission loss, and focuses on the focal plane (Item 36), except for small losses within the metal/AR coating on each optical component. Also, none of the reflectors are subject to chromatic aberration. We note that  
Figure 4: The Y-\(\bar{\text{Y}}\) diagram of the afocal Mersenne design, with the optical stop located at the second, or the negative, lens. The letters P and S stand for the primary and the secondary mirror; and AS denotes the aperture stop, in this case, at the secondary, that is the negative mirror. This figure is identical to the Y-\(\bar{\text{Y}}\) diagram in Figure 2, featuring the Y-\(\bar{\text{Y}}\) diagram of the classical Cooke triplet, without the tertiary mirror and its image. Figure 5: The RT optical design is featured in US Patent 4,240,707 [17], here slightly modified. Item 36 is the focal plane.