Radius and Refractive Index Effects on Lens Action
The action of a simple bi-convex thin lens is governed by the principles of refraction (which is a function of lens curvature radius and refractive index), and can be understood with the aid of a few simple rules about the geometry involved in tracing light rays through the lens. This interactive tutorial explores how variations in the refractive index and radius of a bi-convex lens affect the relationship between the object and the image produced by the lens.
The tutorial initializes with a symmetrical bi-convex thin lens having a default refractive index of 1.6 and a radius of 80 millimeters producing an image of the object (an arrow) positioned 135 millimeters from the lens. To operate the tutorial, use the Lens Radius slider to adjust this value between a range of 60 and 100 millimeters. As the lens radius is increased, the lens grows thinner (less rounded) and the focal length also increases. Note that the blue spherical focal point markers (labeled F for the object space and F' for the image space) move back and forth along the optical axis as the Lens Radius slider is translated. Increasing or decreasing the lens radius affects the size and position of the image formed by the lens. The image Magnification and identity (Real or Virtual) are presented above the image space focal point and are continuously updated as the image size and position change.
The refractive index of the virtual bi-convex lens can be varied between a value of 1.4 and 1.8 by translating the Index of Refraction slider. As the refractive index is increased, the focal points (blue spheres) move closer to the lens, and vice versa. In a manner similar to altering the lens radius, changes in the refractive index influence the size and position of the image. The Focal Length, Object Height, Image Position, and Image Height are presented in the space to the left of the lens, underneath the object space focal point. As the sliders are translated, these values are constantly updated by the tutorial. The Object Position slider (variable between a value of 45 and 145 millimeters) can be employed to adjust the relationship between the object and the lens as the radius and refractive index values are changed.
The object (or specimen) being imaged by the lens is positioned in the object plane, located on the left-hand side of the lens by convention, and is represented by a gray arrow that travels upward from the centerline or optical axis, which passes through the center of the lens, perpendicular to the principal planes. Ray traces through the lens (red and gray lines) emanate from the object and proceed from left to right through the lens to form a real image (inverted gray arrow) in the image plane on the right-hand side of the lens. Characteristic ray traces, including the principal ray, are presented as red lines in the tutorial, while other rays are illustrated as gray traces. The distance between the front principal plane of the lens and the specimen is known as the object distance, and the distance from the rear principal plane to the image is termed the image distance. These parameters are the fundamental elements defining the geometrical optics of a simple lens and can be used to calculate important properties of the lens, including focal length and magnification factor.
Bi-convex lenses are the simplest magnifying lenses, and have a focal point and magnification factor that is dependent upon the curvature angle of the surfaces. Higher angles of curvature lead to shorter focal lengths due to the fact that light waves are refracted at a greater angle with respect to the optical axis of the lens. The symmetric nature of bi-convex lenses minimizes spherical aberration in applications where the image and object are located symmetrically. When a bi-convex optical system is fully symmetric (in effect, a 1:1 magnification), spherical aberration is at a minimum value and coma and distortion are equally minimized or cancelled. Generally, bi-convex lenses perform with minimum aberrations at magnification factors between 0.2x and 5x. Convex lenses are typically utilized for focusing applications and for image magnification.
A lens operates by refracting the incoming light wavefronts at points where they enter and exit the lens surfaces. The angle of refraction, and therefore the focal length, will be dependent upon the geometry of the lens surface as well as the material used to construct the lens. Materials with a high index of refraction will have a shorter focal length than those with lower refractive indices. For example, lenses made of synthetic polymers, such as Lucite (refractive index of 1.47), have a lower refractive index than glass (1.51) leading to a slightly longer focal length. Fortunately, the refractive indices of Lucite and glass are so close together that Lucite can be used in place of glass in many lens applications, including the popular Film-in-a-box cameras that are currently enjoying widespread consumer use. A lens made of pure diamond (refractive index of 2.42) would have a focal length significantly less than either glass or Lucite, although the high cost of pure diamond would be prohibitive for lens construction.