Every optical system, no matter how carefully crafted, deviates from perfection. All optical systems contain Blur. Blur comes from 2 sources, Diffraction and Aberrations. Diffraction is a physical limits because light is found to bend around corners and causes Blur. Aberrations are introduced from imperfections in optics, and even fibers.
In the mid-19th century Ludwig von Seidel formalized a set of five third-order (3rd order and 4th order are the same, and a very unfortunate naming convention because in the Wave Aberration Equation in cartesian coordinates Seidel Aberrations are 3rd order, but in polar form they are 4th order) aberrations that are common among all rotationally symmetric optical systems.
This article will explore these 4th order aberrations from three complementary perspectives:
All plots and figures are created using this downloadable MATLAB script. https://4420950.fs1.hubspotusercontent-na1.net/hubfs/4420950/aberrations_plots.m
The Wave Aberration Function: The Starting Point
Before naming the aberrations, we define the wave aberration function.
In the exit pupil, a perfect wavefront is spherical with radius R :
Aberrations are deviations from this perfect paraboloidal shape introduce extra terms in the function above, which can be expressed in exit pupil coordinates:
This key step from the ideal paraboloidal shape to the full aberration expansion is to recognize that the wavefront can be expressed as a power series in pupil coordinates
The general transverse ray wavefront equation becomes:
In Polar form, which is most commonly used because of the rotational symmetry most optical systems have can be written in terms of normalized pupil coordinates and field coordinates, where:
Expanded gives the most popular form:
This is the most convenient form because each index corresponds to the power of each coordinate, which tells us about why a particular aberration is present in our optical system.
Using the coefficient is an easy way to remember the relationship of the wavefront has in the exit pupil.
Spherical
The Field Coordinate is not present and is therefore constant over the field. This implies this is an on-axis aberration. In the image, rays from the on-axis object point will intersect the axis in front or behind the Gaussian focus. The Wavefront deviation is greater with rays closer to the edges of the pupil and will focus faster. The rays that are closer to the center of the pupil will focus slower. This originates from the lens surface bending light unequally across the pupil.
The Plot on the left is the same plot as the OPD Fan in Zemax and the Plot on the right is the 3D view of the OPD Fan.
Notice the failure to focus. There is a function for focal length radially dependent on where you are in the pupil (here the pupil is the lens itself). Custom operand associated with Spherical is SPHA. Another place to review Spherical in your system is Analyze -> Aberrations -> Seidel Coefficients
Coma
Coma is a failure to magnify off-axis light equally. Magnification varies with field height. This occurs because light from opposite sides of the lens is refracted at different powers, due to the different angles of incidence at the optical surface.
The Blur associated with each concentric circle within spherical can also be seen in coma, however each circle is pushed into a direction that is proportional to the diameter of the circle. This is because unlike Spherical there is a field dependent term. Off-axis point sources produce comet-shaped blurs due to asymmetric magnification across the pupil.
In Zemax, the corresponding operand for coma is COMA.
Astigmatism
Astigmatism is the most difficult aberration to correct. There is a quadratic field dependence, a quadratic pupil dependence. The simplest explanation is that this occurs when the focus of a lens is dependent on the axis of the lens surface. In other words the surface of the lens has different power associated with different axes. Another way of looking at this is that the lens has both a focus and magnification error. It can be an on-axis or off axis aberration, where a single lens has two perpendicular foci, tangential and sagittal rays.
Astigmatism is complicated to view from a ray tracing diagram given the two foci for tangential and sagittal dependence. The pupil term is modulated by the quadratic cosine term, which mathematically gives a different result for the tangential and sagittal planes.
In Zemax, the corresponding operand for Astigmatism is ASTI.
Field Curvature
Field curvature has the same pupil dependence as Defocus, but varies quadratically with field height. This is a failure to focus the entire image on a single plane. Instead, the focal plane is a paraboloid surface resembling a bowl, called the Petzval Surface of Curvature. Even when astigmatism is corrected, the sharp focus for various field points lie on a curved Petzval Surface rather than a flat Image Plane. This aberration is inherent to any lens system with optical power, and needs a field lens for flattening this focus surface.
Field curvature is directly related to astigmatism, however it is not modulated by the cosine term. It is the lens systems inability to focus all fields and rays in the pupil at the same time.
In Zemax, the corresponding operand for Field Curvature is PETC.
Distortion
Distortion is a failure to magnify the image equally across the Field of View. Each ray point in the image is shifted proportional to the cube of the Image location away from the focus. Can be seen as "pincushion or barrel", which is more of a warping of shapes rather than a Blur of points. Corrected using field flatteners.
In Zemax, the corresponding operand for Distortion is DIST.