Melles Griot Optics Guide - Basic Paraxial Formulae

Imaging Properties of a Lens System

Basic Paraxial Formulae

Paraxial theory deals with rays that are sufficiently close to the optical axis that the condition sin q = q is applicable, where q is the angle that a focused or divergent ray makes with the optical axis. Focal Length: Typically, the first step in optical problem solving is to select a system focal length based on constraints such as magnification or conjugate distances (object and image distance). The relationship among focal length, object position, and image position is given by

where f is the focal length of the lens, s is the distance of the object from the primary principal point and s" is the distance of the image from the secondary principal point. Magnification: By definition, magnification (m) is the ratio of image size to object size or

where h is the height of the object and h" is the height of the image. This relationship can be used to recast the first formula into the following forms




where (s + s") is the approximate object-to-image distance. With a real lens of finite thickness, the image distance, object distance, and focal length are all referenced to the principal points, not to the physical center of the lens. By neglecting the distance between the lens' principal points, known as the hiatus, s + s" becomes the object-to-image distance. This simplification, called the thin-lens approximation, can speed up calculation when dealing with simple optical systems. A simple graphical method can also be used to determine paraxial image location and magnification. This graphical approach, illustrated in the figure below, relies on two simple properties of an optical system. First, a ray that enters the system parallel to the optical axis crosses the optical axis at the focal point. Second, a ray that enters the first principal point of the system exits the system from the second principal point parallel to its original direction (i.e., its exit angle with the optical axis is the same as its entrance angle). Note that by using the thin-lens approximation, this second property reduces to the statement that a ray passing through the center of the lens is undeviated.

Graphical method for determining image location and magnification
F-Number: The paraxial calculations used to determine necessary element diameter are based on the concepts of focal ratio (f-number or f/#) and numerical aperture (NA). The f-number is the ratio of the focal length of the lens to its clear aperture (effective diameter) f.

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