Part 1: Singlet lens element

Jump to Part 2: Multi-element lens assembly

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.  

Singlet lens elements are simple and easy to fabricate.  They are used for many simple, not too demanding optical applications such as light collection, image magnification, etc.  We offer a range of singlet lenses as standard products.  You can also use our lens design wizard to design custom singlet lenses.

Lens aberrations and diffraction

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.   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.  (Download a white paper on chromatic aberrations in mobile imaging lenses).

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.

Field curvature

Even in the absence of astigmatism, there is a tendency of optical systems to image better on curved surfaces than on flat planes. This effect is called field curvature. In the presence of astigmatism, this problem is compounded because there are two separate astigmatic focal surfaces that correspond to the tangential and sagittal conjugates. Field curvature varies with the square of field angle or the square of image height. Therefore, by reducing the field angle by one-half, it is possible to reduce the blur from field curvature to a value of 0.25 of its original size.

Coma

In spherical lenses, different parts of the lens surface exhibit different degrees of magnification. This gives rise to an aberration known as coma. Each concentric zone of a lens forms a ring-shaped image called a comatic circle. This causes blurring in the image plane (surface) of off-axis object points. An off-axis object point is not a sharp image point, but it appears as a characteristic comet-like flare. Even if spherical aberration is corrected and the lens brings all rays to a sharp focus on axis, a lens may still exhibit coma off axis. As with spherical aberration, correction can be achieved by using multiple surfaces. Alternatively, a sharper image may be produced by judiciously placing an aperture, or stop, in an optical system to eliminate the more marginal rays.

Geometric distortion

The image field not only may have curvature but may also be distorted. The image of an off-axis point may be formed at a location on this surface other than that predicted by the simple paraxial equations. This distortion is different from coma (where rays from an off-axis point fail to meet perfectly in the image plane). Distortion means that even if a perfect off-axis point image is formed, its location on the image plane is not correct. Furthermore, the amount of distortion usually increases with increasing image height. The effect of this can be seen as two different kinds of distortion: pincushion and barrel. Distortion does not lower system resolution; it simply means that the image shape does not correspond exactly to the shape of the object. Distortion is a separation of the actual image point from the paraxially predicted location on the image plane and can be expressed either as an absolute value or as a percentage of the paraxial image height. It should be apparent that a lens or lens system has opposite types of distortion depending on whether it is used forward or backward. This means that if a lens were used to make a photograph, and then used in reverse to project it, there would be no distortion in the final screen image. Also, perfectly symmetrical optical systems at 1:1 magnification have no distortion or coma.  In a multi-element design it is possible to minimize distortion by using compensating elements and/or aspheric elements.  For ultra-wide field of view lenses,  we can "tailor" the distortion characteristics to maximize the lens performance (see an example on tailored distortion).

Diffraction

Diffraction, a natural property of light arising from its wave nature, poses a fundamental limitation on any optical system. Diffraction is always present, although its effects may be masked if the system has significant aberrations. When an optical system is free from aberrations, its performance is limited solely by diffraction, and it is referred to as diffraction limited. Diffraction at a circular aperture dictates the fundamental limits of performance for circular lenses. The spot size, caused by diffraction, of a circular lens is given as follows:

d = 2.44 l* f/#

where d is the diameter of the focused spot and l is the wavelength of light being focused.  Notice that it is the f-number of the lens, not its absolute diameter that determines this limiting spot size.

Part 2: Multi-element lens assembly