# Brief Introduction of Lens Aberrations

Due to the influence of many factors such as optical design, processing technology and assembling and adjusting technology, it is impossible to form an ideal image of an object of a certain size. The actual image formed by the lens is always different from the ideal image. This difference in image formation is called the aberration of the lens (or imaging optical system).
The aberration is caused by the physical condition (optical characteristic index) of the optical system. In a sense, any optical system has a certain degree of aberration, and theoretically it is impossible to completely eliminate them. The naked eye and other light receivers also have only a certain resolving power, so as long as the aberration value is less than a certain limit, we think that the aberration of the system has been corrected.
First-order aberration theory
In order to establish a satisfactory aberration theory, a simple method is to start with an accurate ray tracing formula (please refer to relevant books) and expand the sine function of each angle into the form of power series according to McLaughlin's theorem, namely sinθ=θ-θ3/3! + θ5/5! - ……。
For small angles, this power series is a rapidly converging series, each term is much smaller than its previous term, which shows that for paraxial rays, due to the small inclination angle, in the case of first-order approximation, all other terms except the first term can be ignored.
Third-order aberration theory
If we use sinθ=θ-θ3/3+ θ5/5! -…,  for all sine functions of angles in the ray tracing formula! the first two items are replaced, then the obtained results, no matter what form of equation, represent the results of the third-order theory, so that the equation can give a fairly accurate explanation of the main aberrations.
In this theory, the aberration caused by any light, that is, the deviation from the path obtained by Gaussian formula, can be expressed by five sums (S1 to S5), which are called sum of Seidel. If there is no defect in the imaging ability of a lens, the sum of these five should all be zero. But no optical system can satisfy all these conditions at the same time. Therefore, conventionally, we consider each sum separately. If one sum is zero, the aberration corresponding to the sum does not exist.
For example, if the Seidel and S1=0 of a known object point on the axis are equal to 0, the spherical aberration of the corresponding image point does not exist. If S2=0, there is no coma. If S3=0, there is no astigmatism. If S4=0, there is no field song. If S5=0, there is no distortion. These aberrations are called five monochromatic aberrations because they exist for any particular color and refractive index. Another kind of aberration is only shown in polychromatic light.
Ball error
After a single wavelength of light emitted from an object point on the optical axis to the lens is imaged, it will no longer converge to the same point on the image side due to the different condensing capabilities of each point on the spherical surface of the lens, but will form a symmetrical diffuse spot centered on the optical axis. This aberration is called spherical aberration.
The size of spherical aberration is related to the position of the object point and the aperture angle of the imaging beam. When the position of the object point is determined, the smaller the aperture angle, the smaller the spherical aberration will be. As the aperture angle increases, the spherical aberration increases in direct proportion to the higher order of the aperture angle. In a photographic lens, if the aperture number is increased by one gear (the aperture is reduced by one gear), the spherical aberration is reduced by half. Therefore, when shooting, as long as the light intensity allows, a smaller aperture should be used to take pictures so as to reduce the influence of spherical aberration.
1. One-sided spherical aberration
One-sided spherical aberration is proportional to the square of the radius of the annulus on the sphere through which light passes.
2. spherical aberration of thin lens
The intersection of the edge ray and the optical axis to the left of the paraxial ray focus is called positive spherical aberration, whereas the opposite is negative spherical aberration. When the lens shape factor q=+0.4 to q=+1.0, the spherical aberration has a minimum value. If the shape of the lens is changed so that the incident angle of the light on the first side is approximately equal to the exit angle of the second side, the edge light will have the smallest deviation. In other words, the equal distribution of the two refractions can minimize the spherical aberration. For the parallel light incident on the crown glass lens, the spherical aberration is the smallest when q=+0.7.
If aspheric surface is used, the spherical aberration of Dan Toujing can be completely eliminated, but this requires that one surface or two circumferential bands of the lens have different curvatures, but the processing of aspheric surface is relatively difficult. Fortunately, the current aspheric surface processing technology has become increasingly mature.
The relation between shape factor and position factor of minimum spherical aberration: q =-2 (n 2-1) p/(n+2)。Where: position factor p=2f/s-1=1-2f/s'; Shape factor q=(r1+r2)/(r1-r2)
Level 5 Ball Error
In the third-order theory, the spherical aberration is proportional to h 2. However, when the value of h is larger, it must be revised again, then sa = ah 2+BH 4. Where: a and b are constants, ah^2 represents the third-order effect and bh^4 represents the fifth-order effect.
The ring radius of the maximum ball difference can be calculated from the above formula, h=0.707h(max). Therefore, in lens design, the ray passing through the 0.707h(max) annulus is always traced to study the spherical aberration.
coma
The second monochromatic aberration in the third-order theory is called coma. An off-axis object point resembles a comet, so it comes from this. Even if the lens corrects the spherical aberration, all rays can be well focused at one point on the axis, but unless the coma is also corrected, a clear image cannot be obtained at the off-axis object point.
An object point outside the optical axis emits a beam of parallel light to the lens. After passing through the optical system, asymmetric diffuse light spots will be formed on the image plane. The shape of the diffuse light spots is comet-shaped, that is, a tail from thin to thick is dragged from the center to the edge. Its head end is bright and clear, and its tail end is wide, dim and fuzzy. The aberration caused by this off-axis beam is called coma. The magnitude of coma is expressed by the degree of asymmetry of the diffuse light spot it forms. The magnitude of coma is related to both aperture and field of view. When shooting, just like spherical aberration, we can reduce the effect of coma on imaging by appropriately reducing the aperture.
Photography generally refers to the blurring caused by spherical aberration and coma as halo. In most cases, the halo of off-axis points is larger than on-axis points. Due to the existence of off-axis aberration, our requirements for off-axis image points should not be higher than on-axis image points, or at most consistent, i.e. both have the same imaging defects. At this time, we call it isohalo imaging. With the increase of relative aperture, the correction of spherical aberration and coma will be more difficult. When using a large aperture lens, the performance of the lens should be known in advance, and the minimum vignetting of that aperture should be noticed. If possible, the optical aperture should be minimized to improve the imaging quality.
If the magnification of light passing through the lens portion is greater than that passing through the center, the coma is positive and vice versa. The relationship between the radius of coma circle and the shape factor and position factor of the lens can be obtained from the third-order theory.
For a single sphere, coma is caused by spherical aberration on the one hand, and the larger the spherical aberration, the larger the coma will be. On the other hand, the coma caused by the refracting sphere is also related to the position of the diaphragm, i.e. the incident angle of the main ray. If the diaphragm is located at the center of the sphere and coincides with the main and auxiliary axes, coma will not occur regardless of the spherical aberration.
A system with neither spherical aberration nor coma is called a non-halo system.
Large coma seriously affects the imaging quality of off-axis points. Therefore, any optical system with a certain aperture size must correct coma well.
The primary coma is proportional to the square of the aperture and the primary square of the field of view. This is why coma occurs when the field of view is very small.
Astigmatism is also an off-axis image basis, which is different from coma. It is an aberration describing the imaging defects of infinite beamlets and is only related to the field of view. Due to the asymmetry of the off-axis beam, the converging point of the meridional beamlet and the converging point of the sagittal beamlet at the off-axis point are at different positions respectively. The aberration corresponding to this phenomenon is called astigmatism. The projection on the optical axis of the distance between the converging point of the meridional beamlet and the converging point of the solitary beamlet is the astigmatism value.
If S3 is not zero, the lens has astigmatism. The blurred image formed by S3 is the light emitted by a certain object point Q with astigmatism. The focal lines of all the light in the vertical and horizontal planes are T and S positions. The two focal lines are perpendicular to their tangential and sagittal planes, respectively. At a certain L position between T and S, it is like a disk and is the smallest blurred circle in this case.
If T and S are determined by the object points in a vast object field far away, their trajectories will form two paraboloids. For any beam, the astigmatism or astigmatism difference is measured by the distance between the two paraboloids through which the main ray passes. On the optical axis, the two paraboloids contact each other, and the astigmatism difference is zero. Outside the axis, this difference increases with the square of the image height. The left side of the T-plane to the S-plane is called the astigmatism.
For thin lenses, astigmatism is approximately proportional to the focal length, but there is little improvement in astigmatism by changing the shape of the lens. However, adding a diaphragm or a Dan Toujing to a doublet lens can improve astigmatism.
When S3=0, t and s coincide with a paraboloid, which is called the Pozner surface.