## 6 Offner Relay (OR)  
This is the final 3M conversion derived from the Cooke triplet described in this work. However, the OR [29] was initiated as a one-to-one relay in mind, i.e., it is a finite conjugate imaging system (point-to-point imaging) instead of infinite conjugate examples described here up to now. Again, we start with the Cooke triplet of Figure 1. We assume that both the first and the third elements have the same power, equal to half of the second negative element. Next, the separation between elements 1 and 2 is identical to that between elements 2 and 3 and is equal to the radius of curvature of the second element, when dealing with a mirror system. This being a one-to-one conjugate system, the object distance is equal to the image distance, which in turn is equal to the radius of curvature of the element 1 or 3. Thus, the optical layout of the redefined Cooke triplet is displayed in Figure 10. Clearly, this ingenious finite conjugate imaging system exhibits perfect object- and image-space symmetry. It has a unit power, that is, no magnification. The Y-Y diagram, corresponding to the Offner relay of Figure 10, is presented in Figure 11. This diagram is similar to the Y-Y diagram of Figure 2, except that the object in this case is at a finite distance. Furthermore, the components and generalized ray on the positive Y-half-plane are symmetrical to those on the negative Y-half-plane. This diagram has more similarity with the Y-Y diagram in Appendix A than to the diagram in Figure 2. It represents a very special case of the diagram in Figure A3. Figure 9: The US Patent 7,616,378 [28] where a concept of a phase plate on a beam splitter (Item 111) is disclosed. A phase plate is inserted into an RT system to remove newly introduced aberrations in the transmission channel, Item 121A. The phase plate is placed on the transmission (second) surface of the beam splitter. The imaging system produces an image with a unit magnification of the object in the image location. Furthermore, only a small object is being imaged at the edge of the object field. It is effectively reimaged to the other edge of the object. The OR is an optical system that inverts the object and image onto itself. Thus, the chief ray \(\bar{\Upsilon}\) remains practically unchanged, until the narrow bundle of rays is incident on the (top outer) segment of the primary imaging component. The generalized ray remains practically parallel to the \(Y\)-axis because the ray bundle expands insignificantly. The convergent primary component deviates the generalized ray in the clockwise direction toward the small secondary lens or mirror that also functions as the aperture stop. The secondary lens or mirror is divergent; therefore, it deviates the generalized ray in a counterclockwise direction. Afterward, the graph is symmetrical, because both the primary lens or mirror and the secondary one form a part of the same large component and the system features unit magnification. The object and the image are of the same size, O-OA is equal to OA-I. The power of the primary is equal to the power of the tertiary lens or mirror. OA-AS is the radius or half side of a square aperture. A working conversion of the optical layout in Figure 10 or Figure 11 into the all-reflective configuration is shown in Figure 12. Both imaging component 1 and component 3 have the same power, the same separation from the second component and share the same optical axis. For this reason, they may be perceived and fabricated as a single component. This new single component and the second component are concentrically surrounding a shared and common focus. Examining either Figure 11 or Figure 12, we realize that OR is  
Figure 11: Y-\(\bar{Y}\) diagram of the Offner relay, presented in Figure 10, is clearly a one-to-one, finite conjugate system, symmetric with respect to the optical stop.