## Appendix A The Y-Y Diagram
Figure A3a includes the Y-\(\bar{Y}\) diagram for the optical system of Figure 1, working at finite conjugates. The object is located at Point O. The size of the object is O-OA. A generalized ray travels toward the primary P, a convergent lens whose function is to change the graph direction in a clockwise direction until the secondary lens is encountered. The secondary lens S changes the direction of the generalized ray into a counterclockwise direction because it is divergent. The tertiary lens T is convergent again, so it changes the direction of the generalized ray into the clockwise direction. When the generalized ray crosses the \(\bar{\rm Y}\)-axis, at I, the height of the Y-ray is zero and the designer has found the image location. The image size is OA-I. We indicated the units below the diagram to appreciate a demagnification by a factor of 3. As we can see in Figure A3b, the most critical system parameters, the object and the image size, and the location of the aperture stop size may remain fixed even when we change the optical elements that manipulate the generalized rays. This flexibility and clarity make the Y-\(\bar{\rm Y}\) diagram an ideal first-order design tool. Both axes measure distance in [m] or [mm], but we use a different scale on each axis. Aperture diameter is usually measured in tens of cm to a meter, while the image is captured on a focal plane array with size measured in millimeters, and each pixel in microns. The aperture is usually significantly larger than the image, so a different scale is used for both axes. The aperture size (radius, or half-side of a rectangle) is OA-AS. A generalized ray propagates in the same direction in an isotropic medium, such as air where most optical systems operate except for the immersion subsystems. A sharp change in the direction of the graph indicates the presence of a component with power, a lens, or a mirror. A convergent optical element changes the direction of the generalized ray clockwise in the positive Y-half-plane, while a divergent one changes the direction of the ray line in the counterclockwise direction. The stronger the power of the optical elements, the greater the deviation in the line direction. Rays are represented as line segments between optical components and critical planes in the (traditional) optical system. The function of the positive or convergent lens or a mirror is to deviate the generalized ray in a clockwise direction when a component is in the positive Y-plane. The function of a negative or divergent lens or a mirror is to deviate the generalized ray in a counterclockwise direction when the component is in the positive Y-plane. The deviations of the generalized ray are the opposite when the components are in the negative Y-plane. We consider two special cases of ray orientation. The generalized ray is parallel to the \(\bar{\rm Y}\)-axis when the chief ray \(\bar{\rm Y}\) remains at the same height above the optical axis, that is, in the region of afocal ray bundles, or when the object or image is located at infinity. The generalized ray is parallel to the \(Y\)-axis when the ray bundle expands imperceptibly, that is, the slope of the marginal ray is nearly zero.