Useful Formulas Paraxial Optics: Sign Rules: The (local) origin is at the intersection of the surface and the z-axis. All distances are measured from the origin: Right and Up are positive. All distances are directed and add as such. All angles are acute. They are measured from: From the z-axis to the ray From the surface normal to the ray CCW is positive, CW is negative Indices of refraction are positive for rays going left-to-right; negative for rays going right-to-left. (Normal is left-to-right.) Paraxial Assumptions: Surface sag is ignored as negligible. angles sines tangents NOTE: are considered to be tangents of the ray If “angle variables” (e.g., ui, ui’) angles, then Gaussian ray tracing works for finite apertures and represents “perfect” optical surfaces/lenses. Notation Conventions: Unprimed values are before the surface (or lens/system). Primed values are after passing the surface (or lens/system). Some variables have only one (unprimed) value: Curvature, c, and power, K Distance, dn is from the nth surface to the (n+1)th surface or lens. Paraxial Ray Tracing Equations: Refract at a surface, with index n before refraction and after n’ Refract through a thin lens in air Transfer between surfaces separated by distance d Paraxial Imaging Equations: Image location from a surface Image location from a thin lens in air Image to next lens’ object transfer Graphical Ray Tracing Through Paraxial lenses or surfaces Incident rays parallel to the axis, pass through the focal point. Rays that pass through the focal point emerge parallel to the axis. Rays through the center of the lens are undeviated.
Useful Formulas Paraxial Optics: Sign Rules: The (local) origin is at the intersection of the surface and the z-axis. All distances are measured from the origin: Right and Up are positive. All distances are directed and add as such. All angles are acute. They are measured from: From the z-axis to the ray From the surface normal to the ray CCW is positive, CW is negative Indices of refraction are positive for rays going left-to-right; negative for rays going right-to-left. (Normal is left-to-right.) Paraxial Assumptions: Surface sag is ignored as negligible. angles sines tangents NOTE: If “angle variables” (e.g., ui, ui’) are considered to be tangents of the ray angles, then Gaussian ray tracing works for finite apertures and represents “perfect” optical surfaces/lenses. Notation Conventions: Unprimed values are before the surface (or lens/system). Primed values are after passing the surface (or lens/system). Some variables have only one (unprimed) value: Curvature, c, and power, K Distance, dn is from the nth surface to the (n+1)th surface or lens. Paraxial Ray Tracing Equations: Refract at a surface, with index n before refraction and n’ after Refract through a thin lens in air Transfer between surfaces separated by distance d Paraxial Imaging Equations: Image location from a surface Image location from a thin lens in air Image to next lens’ object transfer Graphical Ray Tracing Through Paraxial lenses or surfaces Incident rays parallel to the axis, pass through the focal point. Rays that pass through the focal point emerge parallel to the axis. Rays through the center of the lens are undeviated.