Theories and Formulas
Ray Tracing
By propagating the rays linearly through homogeneous media and applying Snell’s Law at the interfaces between media, light rays can be traced through an optical system. Such a system may consist of various optical components (lenses, prisms and mirrors). Each optical component can be treated as a series of individual interfaces and transfers. For example, a simple lens is a piece of transparent material with two surfaces of specified radii and a thickness.
The ray trace through the lens consists of three segments: refraction at the first surface, propagation through the lens, and refraction at the second surface. For a specific optical system, this series of three simple operations can be repeated to trace the rays through the entire system.
Focal Length
Rays that pass through either focal point are parallel to the optical axis on the other side of the lens. The focal length is related to the object and image locations by the “lens formula”:
1/f = 1/S +1/S′ where S = object distance and
S′ = image distance.
Magnification
The magnification, M, is defined as the ratio of the image height, y′, to the object height, y:
M = y′/y; also, M = S′/S.
Thin Lens Theory
The simplest case is that of a “thin” lens of zero thickness in air. All of the refraction occurs at the plane of the lens, and all distances can be measured from this plane, as shown in Figure 4. The lens is described by
1/f = (n-1) (1/R1 - 1/R2), where f = focal length; R1= radius of curvature of the first surface of the lens; R2 = radius of curvature of the second surface of the lens; n= index of refraction of the material of the lens.
Figure 4
Thick Lens Theory
While a thin lens is a convenient idealization, a physical lens possesses a finite thickness. Accounting for this leads to the “lens maker’s formula” for a thick lens in air:
where t = center thickness of the lens
f = effective focal length (EFL)
R1 = first radius of curvature
R2 = second radius of curvature
n = index of refraction of the lens
