“Aberrations” is a general category including the principal factors that cause an optical system to perform differently than the ideal case. There are a number of factors that do not allow a lens to achieve its theoretical performance.

The homogeneity of optical materials and surfaces is the first requirement to achieve optimum focusing of light rays and proper image formation. Obviously, homogeneity of real materials has an upper limit determined by various factors (e.g. material inclusions), some of which cannot be eliminated. Dust and dirt are external factors that certainly degrade a lens performance and should thus be avoided as much as possible.

Spherical lenses are very common because they are relatively easy to manufacture. However, the spherical shape is not ideal for perfect imaging - in fact, collimated rays entering the lens at different distances from the optical axis will converge to different points, causing an overall loss of focus. Like many optical aberrations, the blur effect increases towards the edge of the lens.

To reduce the problem, aspherical lenses (Fig. 16) are often used - their surface profile is not a portion of a sphere or cylinder, but rather a more complex profile apt to minimize spherical aberrations. An alternative solution is working at high F/#’s, so that rays entering the lens far from the optical axis and causing spherical aberration cannot reach the sensor.

Lens with spherical aberration.

Aspherical lens.

The refractive index of a material is a number that describes the scattering angle of light passing through it – essentially how much rays are bent or refracted - and it is a function of the wavelength of light. As white light enters a lens, each wavelength takes a slightly different path. This phenomenon is called dispersion and produces the splitting of white light into its spectral components, causing chromatic aberration. The effect is minimal at the center of the optics, growing towards the edges.

Chromatic aberration causes color fringes to appear across the image, resulting in blurred edges that make it impossible to correctly image object features. While an achromatic doublet can be used to reduce this kind of aberration, a simple solution when no color information is needed is using monochrome light. Chromatic aberration can be of two types: longitudinal (Fig. 17) and lateral (Fig. 18), depending on the direction of incoming parallel rays.

Longitudinal/axial chromatic aberration.

Lateral /transverse chromatic aberration.

Astigmatism is an optical aberration that occurs when rays lying in two perpendicular planes on the optical axis have different foci. This causes blur in one direction that is absent in the other direction. If we focus the sensor for the sagittal plane, we see circles become ellipses in the tangential direction and vice versa.

Astigmatism aberration.

Coma aberration occurs when parallel rays entering the lens at a certain angle are brought to focus at different positions, depending on their distance from the optical axis. A circle in the object plane will appear in the image as a comet-shaped element, which gives the name to this particular aberration effect.

Coma aberration

Field curvature aberration describes the fact that parallel rays reaching the lens from different directions do not focus on a plane, but rather on a curved surface.

This causes radial defocusing, i.e. for a given sensor position, only a circular crown will be in focus.

Field curvature aberration.

With a perfect lens, a squared element would only be transformed in size, without affecting its geometric properties. Conversely, a real lens always introduces some geometric distortion, mostly radially symmetric (as a reflection of the radial symmetry of the optics). This radial distortion can be of two kinds: barrel and pincushion distortion. With barrel distortion, image magnification decreases with the distance from the optical axis, giving the apparent effect of the image being wrapped around a sphere. With pincushion distortion image magnification increases with the distance from the optical axis. Lines that do not pass through the center of the image are bent inwards, like the edges of a pincushion.

Distortion

Since telecentric lenses are a real-world object, they show some residual distortion which can affect measurement accuracy. Distortion is calculated as the percent difference between the real and expected image height and can be approximated by a second-order polynomial. If we define the radial distances from the image center as follows

`R_a = actual Radius`

`R_e = expected Radius`

the distortion is computed as a function of Ra:

`dist (R_a) = (R_a - R_e) / R_a = c * R_a^2 + b * R_a + a`

where a, b and c are constant values that define the distortion curve behavior; note that “a” is usually zero as the distortion is usually zero at the image center. In some cases, a third-order polynomial could be required to get a perfect fit for the curve.

In addition to radial distortion, also trapezoidal distortion must be taken into account. This effect can be thought of as the perspective error due to the misalignment between optical and mechanical components, whose consequence is to transform parallel lines in object space into convergent (or divergent) lines in image space.

Such an effect, also known as “keystone” or “thin prism”, can be easily fixed by means of pretty common algorithms which compute the point where convergent bundles of lines cross each other.

An interesting aspect is that radial and trapezoidal distortion are two completely different physical phenomena, hence they can be mathematically corrected by means of two independent space transform functions that can also be applied subsequently.

An alternative (or additional) approach is to correct both distortions locally and at once: the image of a grid pattern is used to define the distortion error amount and its orientation zone by zone. The final result is a vector field where each vector associated with a specific image zone defines what correction has to be applied to the x,y coordinate measurements within the image range.

All lenses operating in the visible range, including Opto Engineering® telecentric lenses, are achromatized through the whole VIS spectrum. However, parameters related to the lens distortion and telecentricity are typically optimized for the wavelengths at the center of the VIS range, which is green light. Moreover, the resolution tends to be better in the green light range, where the achromatization is almost perfect. “Green” is also better than “Red” because a shorter wavelength range increases the diffraction limit of the lens and the maximum achievable resolution.