Lens Bending - Is it really a good thing???
Right! So I have mentioned lens bending and its effect on aberrations. Is this what I was talking about?
Well, no. Obviously not. Lens designers bend rays, not lenses! Or do they?
Lens element bending does have a profound effect on aberrations, and if combined properly, multiple bent lens elements can balance the aberration content. Different optical glasses such as crowns, flints, and short flints can be used in combination with element bending to provide overall balance, and is the basis on which all multielement lenses are based. So let's take a look first at some simple aberrations so that we can better appreciate the impact of this bending effect. One of the best overall texts on aberrations is arguably W T Welford's Aberrations of Optical Systems published by Adam Hilger. However, it is very easy to get lost in the math presented by Welford. Therefore, one of the best and basic texts, in my opinion, is Warren Smith's Modern Optical Engineering published by McGraw Hill. Many of the lens bending and stop shift diagrams shown below are from Chapter 3 of this text - a text that I can highly recommend.
Aberrations can be described mathematically quite accurately by an aberration polynomial that contains third order, fifth order, seventh order, etc. terms. Each order becoming more complex in the mathematics used to describe the aberration behavior. When you see "third order" here, it describes the most basic and the most significant element of the particular aberration. In addition to the basic aberrations of spherical, coma, astigmatism, distortion, chromatic, we control the performance of a lens by shifting the stop relative to the elements.
Spherical Aberration (third order)
Spherical Aberration describes a very common behavior in which the focal length of the lens varies with position of the ray within the lens aperture. This is illustrated well in the figure below.
This diagram illustrates "under-corrected third order spherical aberration" in which the rays at larger aperture heights focus shorter than those near the axis. Over-corrected would describe that these rays would focus longer than those on axis. In imagery, spherical looks like a defocused image, but while shifting the focal plane axially will cause defocus to disappear at some position, spherical defies this and never allows a truly sharp focus.
Coma (third order)
Coma is an "off-axis" aberration, meaning that it manifests itself only in non-axial object points. It has a very characteristic appearance in imagery, as depicted below, especially if bright points occur out in the field of view.
The cause of this aberration can be better understood by considering the ray bundle behavior that causes it. As shown in the diagram below, rays transmitted by the lens farther from the axis come to focus at a distance farther from the axis than those transmitted closer to the lens axis.
Distortion (third order)
Distortion can be described as a change in magnification as a function of field position. The diagram below depicts the two "forms" of distortion, "pincushion" or positive distortion on the left, and "barrel" or negative distortion on the right.
Astigmatism results from the existence of two different focal lengths in two different meridians. A diagram is useful in understanding how this occurs. Rays emitted from an off-axis object point strike the surface of a spherical lens with a multiplicity of incidence angles. For the moment, consider only those that strike the lens surface in two orthogonal planes; one containing the off-axis object point and the optical axis, the second, at right angles to it and containing the same chief ray (refer to the previous blog for the definition of the chief ray). The first plane is referred to as the tangential plane, the second as the sagittal plane. Astigmatism occurs when the shape of the (perfect spherical) lens causes the rays in each of these two planes (fans of rays) to focus in two different planes positioned along the optical axis due to the difference in the angles of incidence at each surface. These foci appear as lines that are at right angles to one another, but displaced along the optical axis.
Note that the heading uses the plural form; there are two similar but different forms of chromatic aberration that result from the dispersion of glass materials. The spectral separation of white light (dispersion) by a prism or the water droplets in air is recognized and enjoyed by nearly everyone. This is because the velocity of blue light (shorter wavelengths) in these higher index materials is reduced more than red light (longer wavelengths), resulting in a shorter focal length for blue light than for red light.
Longitudinal Chromatic Aberration
The first chromatic effect can be expressed as two different measurements: longitudinal color or transverse color depending on how you wish to describe the anomoly. This chromatic aberration occurs on axis, and is therefore independent of stop position.
Lateral Chromatic Aberration
Additionally, we observe a second chromatic effect, "lateral chromatic aberration" that can be thought of as a change of magnification with spectrum when the aperture stop (we discussed this in the previous blog) is shifted from the lens, and we consider objects or edges in the field of the lens. As we move farther out in the field, we observe a greater separation of blue and red (as well as all other colors of course). The "red image" is larger than the "blue image".
As seen in the Lateral Chromatic Aberration case above, we can readily see that bundles of rays from a given object point are most often NOT evenly distributed about the lens element axis due to the position of the stop relative to the lens element. Therefore the "blend" of aberrations changes with the position of the stop relative to the individual element. So let's start the process of looking at lens bending effects on abberations in concert with stop position with two of the "less complex" third order aberrations: spherical and coma. (Recall from the previous blog that, by convention, light ALWAYS travels from left to right, and in the diagrams that follow, object space is to the left, image space is to the right of the lenses depivted at the bottom.)
What we see in the plot immediately above is that coma and spherical can be minimized at very nearly the same bending of a positive lens, but they cannot be simultaneously eliminated. (Spherical cannot be eliminated completely regardless of bending.)
We could spend a LOT of time on the details of aberrations, but part of the reason for addressing this subject is to familiarize the reader with the "simpler" and more significant forms of the aberrations and to recognize that changing the lens shape to minimize one aberration can, and usually does, aggravate another. When we add stop shifting to the mix, we begin to see that the relationship is extremely complex for a single element, and that trying to mathematically optimize the performance of a lens of 10-20 elements by using only the aberration polynomials and stop shift equations would be a gargantuan task. However, these highly non-linear relationships can be used to counter one another, but the way we do it is to use a digital computer with a very fast processor, we define a merit function that describes performance that we desire, and we trace many, many rays and allow the optimization algorithm to find a minimum to the merit function that provides acceptable performance.
Experience in assessing the current performance and how the lens performance is changing under the "rules" defined by the merit function allows a lens designer to alter the merit function during the course of optimization and "steer" the optimization process in a desired direction. Hence, there is no "silver bullet" merit function that works every time. The value of a lens designer is in knowing what lens form to start with, and how to "steer" the optimization process to attain an optimum result (which is never perfect). One might add constraints on this statement such as "at reasonable cost", "in a manufacturable form", or "with available glasses". And by the way, there are literally hundreds of high quality optical glasses with various indices, dispersions, partial dispersions, spectral transmissions, and manufacturing parameters and costs to choose, from at least a dozen different reputable glass manufacturers.
The SMA website has a number of lens types described along with some case studies of lenses that we have designed. Take a look around. Feel free to leave me a comment on the "Contact Us" page.
Next time we will shift gears a little and consider the manufacturing and tooling trade-offs of plastic versus glass optics.