When collimated, on-axis light is focused by an ideal lens. All light rays cross the optical axis at a single point, forming a spot with a diameter determined by the diffraction formula in the Spot Size tutorial. However, many lenses exhibit a phenomenon termed spherical aberration. This causes light rays impinging near the lens edge to cross the optical axis closer to the lens than those going through the lens center, as shown in the graphic at left.
Spherical aberration increases spot size and causes best focus to occur at a different location than the calculated effective focal length.
Spherical aberration, a function of several factors, includes lens shape, orientation, and index of refraction. For example, the best shape for a crown glass lens used to focus visible light to a minimum spot size is a biconvex lens. Conversely, for a ZnSe lens used at 10.6µm, the best design for a minimum spot size is a meniscus lens.
The exact spot size for a given lens under specific circumstances is determined by ray tracing; however, a useful formula for estimating the spot size due to spherical aberration in a best form lens is:
- f is lens focal length
- D is input beam diameter at the lens (at the 1/e2 point)
- k is an index of refraction function
The most important point to note from the preceding formula is that the spot size due to spherical aberration is proportional to the cube of the beam diameter and inversely proportional to the square of the focal length. Thus, as the laser beam diameter decreases for a given lens, spot size rapidly decreases due to spherical aberration. Similarly, as focal length increases for a given laser beam diameter, the spherical aberration spot size is again reduced. For all the materials listed, the k value is significantly smaller for meniscus lenses than for plano-convex lenses. Thus, when spherical aberration is significant, the meniscus lens will perform better than the plano-convex lens.
The value of k is given for several materials at 10.6µm in the following table:
Spherical Aberration Table