Amazingly, this demonstrations of tracing rays though lenses also supports the theory of rendering images with ray tracing. The equations for image ray tracing are the same as those used in this java applet. 
 
 

• Characteristics of non-thin lenses 
• Other lenses demonstrated by ray tracing applet 
• Ray Tracer Java applet 

Theory matched to demonstration

A lens is defined it's focal length, which is the distance from the lens that lights rays from an infinite far away object converge to after passing though the lens. Infinite means that the light rays coming from a point on an object can be considered parallel. The equation for the focal length is (F = focal length; R1 = radius left side; R2 = radius right side;  n = refractive index of incident material) : 
 
 

The example of the bi-convex and bi-concave lenses verify this (see Figure 1). The light ray in the diagram below converge at the focal length. The radius of the lenses is measured in units of the grid (refractive index of glass 1.33). 

Figure 1: Light rays converge at focal length (count grid mark to verify). (a) bi-convex (b) bi-concave. 

The theory of thin lenses predicts the magnification from object to image. A bi-convex lens has magnification x(-1) when the object (point light source) is 2F away from the lens (see Figure 1). In addition, a virtual image is created when the object is closer than F to the lens (seen by the exiting light diverging from each other (extend the rays back). This is seen in the Figure 2. 

Figure 2: Bi-convex lens, Magnification x(-1) at object 2F from lens (count grid marks to verify) 

A bi-concave lens only forms virtual images that are smaller than the object. The black light rays are extended forward and backward to illustrate the convergence to the virtual image. 

Figure 3. Bi-concave lens, virtual image is smaller than the object. 
 

Verified features of a thick lens

The demonstration of the applet also shows characteristics of lenses found in real lenses. Light rays traveling through "real/thick" lenses are influenced aperture size, depth-of-field, ray wavelength, and curvature of lens. Most of the examples are for bi-convex lenses. The ideas extend to other lens also. 

Some definition of aperture size and depth-of-field before continuing. The apperture size is the area of the lens through which the light rays pass though. Depth-of-field is the range of distance an object is away from the lens, so that the formed image is considered in focus. In other words, the point of focus changes with distance to object. The depth-of-field effect refers to the blurred image when the object moves out of focus. This is demonstrated in Figure 4, where the bunny is in focus and the closer and distant penguin are out of focus. The depth-of-field is from the lens to the bunny. Figure 5 illustrates depth-of-field for two point sources. The formed images converge at different distances from the lens. One image would be considered blurred relative to the other as they are not on the same plane. The blurred image would be called "circle of confusion". 

Figure 4: Depth-of-field effect. click image to see source. 

Figure 5: Depth of field for light rays. 

The aperture size will influence the effect depth of field. The point of focus is clearest for rays coming through the lens center (aperture size is smallest). Light rays passing further from the center (larger aperture) diverge more from the point of focus (see Figure 6). This is probably from the fact that a small change in angle leads to a big change at a destination further away.  The deviation might be corrected by using parabolic shaped lenses instead of spherical ones. 

Figure 6. Smaller aperture more focused rays. 

Similarly, different wavelength of light will slightly blur an image. Light rays are refracted at different angles due to wavelength. Figure 7 shows how each wavelength converges at different points. Figure 7a is very messed, but 7b shows the green rays removed, and the separation between red and blue is obvious. This light spread is a prism effect. The refractive indexes: r=1.3; G=1.4;B=1.6; 
 
 

Figure 7. (a) RGB rays turned on. (b) RB rays on, clearer separation. 
 

Figure 8. Convex-plan lens. See longer focal length than bi-convex lens in Figure 1. 

In addition, the lens tracer applet can illustrate light tracing through several lenses as seen in Figure 9. The convex lenses converge the light where by the concave lenses diverge the light rays. 

Figure 9. Several lenses back to back.