## Appendix A The Y-Y Diagram  
The name Y-Y diagram arises from the names of two important rays in the first-order optical system design. A designer uses the first-order design when they choose a focal length (power) of a lens, knowing the object and the image distance. In the formal analysis of optical systems, the first-order design, or the analysis applicable in the paraxial region, is characterized by the possibility of replacing tangents and sines of the angles with the angles themselves. Consequently, this analysis applies only in a narrow region close to the optical axis. Furthermore, the imaging optical components such as lenses and mirrors are represented as infinite planes normal to the optical axis. Being visually oriented, we often insert a convex or concave lens as a symbol. The object and the image are also located on such a plane. Each point R on the graph of a generalized ray on the Y-Y diagram has mutually independent coordinates (Y\(\widehat{Y}\)). We can think of it as a vector. The only thing that connects the points on this graph is that the next point on the graph is determined by the coordinate of the previous point and the paraxial ray trace equations, namely, transfer and refraction at the surfaces with power. Object height, measured from the optical axis, has traditionally been set to be unitary, to allow scaling in the traditional (hand) ray-trace calculations. The image height is obviously dependent on the specifics of the optical system. A marginal ray is a ray that originates at the object's on-axis point and is directed to the edge of the aperture stop. Its height at the stop location determines the radius of the aperture stop or one-half of the side of the square aperture. It is denoted as Y; sometimes it includes a subscript that specifies the position along the optical, or \(z\)-axis, at which it is being evaluated. Every point where the marginal ray crosses the optical axis is the image location. The travel of the marginal ray through the finite conjugate optical system of Figure 1 after converting it into finite conjugates is illustrated in Figure A1. Chief ray originates at the bottom object point and is directed to the center of the aperture stop, i.e., it crosses the optical axis at the stop location. An optical system has a single aperture stop, which defines its equivalent optical system, the f-number, and its throughput. The height of the chief ray at the image location determines the image height, including for the intermediate images. It is denoted as \(\bar{Y}\); sometimes it includes a subscript that determines the position along the optical, or \(z\)-axis, at which it is being evaluated. Similarly, Figure A2 illustrates the travel of the chief ray through the optical system of Figure 1, working at finite conjugates. Sometimes a designer is interested in how the imaging components (lenses, mirrors) impact the primary parameters of the optical system such as object and image height, stop location, and size. For the applications where the distances along the length of the optical system are not of significance, we can present graphically how the chief and the marginal ray evolve through the optical system. The values of chief and marginal ray are known at each point along the optical system, upon ray-trace analysis. Figures A1 and A2 present these rays propagating along the z- or optical axis. In the Y-\(\bar{Y}\) diagram, the \(z\)-axis is facing into the page, \(\bar{Y}\)-axis is drawn along the horizontal axis, and \(Y\)-axis is vertical. All three axes are mutually perpendicular. The construction of these two rays represents a rigorous solution to the wave equation. This is a second-order differential equation that requires two independent (orthonormal) solutions. We can consider the trajectory of the generalized ray inside the Y- and \(\bar{Y}\)-diagram to be a vector (\(\bar{Y}\), \(\bar{Y}\)), dependent implicitly on the \(z\)- (optical) axis. A point in the Y-\(\bar{Y}\) plane corresponds to a z-value (a point on the axis of the optical system).