Singlet lens elements

Singlet lens elements are the building blocks of all lens systems. There are five basic lens element shapes: plano-convex, bi-convex, plano-concave, bi-concave and meniscus.  Plano-convex and bi-convex lenses have positive power.  They will converge an parallel input beam into a real focal point at some distance behind the lens elements.  Plano-concave and bi-concave elements have negative power.  They will diverge an parallel input beam from a virtual point in front of the lens elements. A meniscus lens can be either positive or negative depending on the two surface curvature and the thickness of the element.  The optical function of lens element is shown as follows:

An ideal lens will focus an input parallel beam to a perfect point (focal point).  The size of the focal point should be infinitesimal. However, because of lens aberrations and diffraction,  the focal spot of a real lens has a finite size. The size of the focal spot is a measure of lens aberrations and diffraction.  All singlet lenses have significant amount of aberrations.

Spherical aberration

Take an parallel input ray bundle on-axis,  a perfect lens will focus this bundle to an infinitesimal point.  However a real lens cannot do this.  The light distribution near the focal point is shown as follows. Rays near the edge of the bundle will focus somewhere closer to the lens than rays closer to the axis (paraxial rays). The distance between the paraxial focal point and the edge ray focal point is known as longitudinal spherical aberration. When the edge rays intercept the paraxial focal plane, the interception points are also displaced from the paraxial point.  This separation is known as transverse spherical aberration. The amount of spherical aberration depends on several factors. Among them are: lens shape and material index.  Spherical aberration decreases with index of refraction. For a given index of refraction, it is possible to design a singlet lens element free of 3rd order spherical aberration.  The resulting lens shape is known as "best form" singlet.  A design wizard is provided that will design "best form" singlet lenses based on your input parameters.  It is also possible to eliminate spherical aberration completely if the surface of the lens element can be made aspherical.  Most laser pointer lenses are designed and made this way.  However due to manufacturing limitations, it is extremely expensive to make aspherical elements with glass materials. It is much more economical to make aspherical elements with molded plastic materials.

Chromatic aberration

The aberrations previously described are purely a function of the shape of the lens surfaces, and can be observed with monochromatic light. There are, however, other aberrations that arise when these optics are used to transform light containing multiple wavelengths. The index of refraction of a material is a function of wavelength. Known as dispersion, this is represented by the Abbe value of the material. The following figure shows the result when polychromatic collimated light is incident on a positive lens element. Because the index of refraction is higher for shorter wavelengths, these are focused closer to the lens than the longer wavelengths. Longitudinal chromatic aberration is defined as the axial distance from the nearest to the farthest focal point. Lateral color is the difference in image height between blue and red rays. Because of the change in index with wavelength, blue light is refracted more strongly than red light, which is why rays intercept the image plane at different heights. Stated simply, magnification depends on color. Lateral color is very dependent on system stop location. For many optical systems, the third-order term is all that may be needed to quantify aberrations. However, in highly corrected systems or in those having large apertures or a large angular field of view, third-order theory is inadequate. In these cases, exact ray tracing is absolutely essential. As in the case of spherical aberration, positive and negative elements have opposite signs of chromatic aberration. By combining elements of nearly opposite aberration to form a doublet, chromatic aberration can be partially corrected. It is necessary to use two glasses with different dispersion characteristics, so that the weaker negative element can balance the aberration of the stronger, positive element. Achromatic doublets are superior to simple lenses because achromatic doublets correct for spherical as well as chromatic aberration, they are often superior to simple lenses for focusing collimated light or collimating point sources, even in purely monochromatic light.

Astigmatism

When an off-axis object is focused by a spherical lens, the natural asymmetry leads to astigmatism. The system appears to have two different focal lengths. As shown below, the plane containing both optical axis and object point is called the tangential plane. Rays that lie in this plane are called tangential rays. Rays not in this plane are referred to as skew rays. The chief, or principal, ray goes from the object point through the center of the aperture of the lens system. The plane perpendicular to the tangential plane that contains the principal ray is called the sagittal or radial plane. The figure illustrates that tangential rays from the object come to a focus closer to the lens than do rays in the sagittal plane. When the image is evaluated at the tangential conjugate, we see a line in the sagittal direction. A line in the tangential direction is formed at the sagittal conjugate. Between these conjugates, the image is either an elliptical or a circular blur. Astigmatism is defined as the separation of these conjugates. The amount of astigmatism in a lens depends on lens shape only when there is an aperture in the system that is not in contact with the lens itself. (In all optical systems there is an aperture or stop, although in many cases it is simply the clear aperture of the lens element itself.) Astigmatism strongly depends on the conjugate ratio.