Telescope coma aberration is a prevalent issue in optical systems, particularly in widefield telescopes and fast lenses. This aberration occurs when light rays from a point source at the edge of the field of view are not focused to a point, resulting in a “cometlike” shape with a tail pointing away from the center of the image. Understanding and quantifying this phenomenon is crucial for designing and optimizing optical systems.
Understanding Coma Aberration
Coma aberration is a type of thirdorder aberration that arises due to the nonuniform magnification of the image across the field of view. This nonuniform magnification is caused by the difference in the optical path lengths between the center and the edge of the lens or mirror. As a result, the image of a point source at the edge of the field of view appears distorted, with a characteristic “cometlike” shape.
The level of coma aberration in an optical system can be described using Zernike polynomials, which are a set of orthogonal functions used to represent the wavefront aberrations. The Zernike coefficient for coma aberration is denoted as C₃₁, and it can be calculated using the following formula:
C₃₁ = (8/3)√(3/5) × (h/f)³
Where:
– h is the distance of the point source from the optical axis
– f is the focal length of the lens or mirror
The value of C₃₁ represents the magnitude of the coma aberration, with higher values indicating more severe distortion.
Quantifying Coma Aberration
To quantify the level of coma aberration in a telescope, a simple scale can be used. This scale measures the width of the aberration at its longest point in pixels and compares that dimension as a percentage of the height of the image. The following guidelines can be used to interpret the level of coma aberration:
Aberration Level  Percentage of Image Height 

Excellent  0.4% or less 
Acceptable  Less than 1% 
Poor  1% or higher 
For example, if the width of the coma aberration is 10 pixels and the image height is 2000 pixels, the aberration level would be 0.5%, which is considered “acceptable” performance.
Practical Example: Canon EF 50mm f/1.8 STM Lens
To illustrate the concept of telescope coma aberration numericals, let’s consider the example of the Canon EF 50mm f/1.8 STM lens. This lens is a popular choice for astrophotography due to its relatively fast aperture and affordable price.
At the maximum aperture of f/1.8, the lens shows very poor performance in the corner of the fullframe image, with aberration levels reaching 1.5%. This means that the width of the coma aberration is 1.5% of the image height, which is considered “poor” performance.
However, by stopping down the lens to f/2.8, the level of coma aberration is significantly reduced, reaching a very good level of 0.5%. This is consistent with how the lens is typically used for astrophotography, where the aperture is often stopped down to improve image quality and reduce aberrations.
Theoretical Explanation and Physics
The theoretical explanation for coma aberration can be found in the principles of geometric optics and the Seidel aberration theory. Coma aberration is a result of the nonuniform magnification of the image across the field of view, which is caused by the difference in the optical path lengths between the center and the edge of the lens or mirror.
Mathematically, coma aberration can be described using Zernike polynomials, as mentioned earlier. The Zernike coefficient for coma aberration, C₃₁, represents the magnitude of the aberration and can be used to predict the shape and size of the distortion.
The level of correction for coma aberration depends on the sample structure and the specific design of the optical system. In some cases, the aberration may be corrected using specialized lens elements or mirror coatings, while in other cases, the aberration may be minimized by stopping down the aperture or using a larger focal length.
Numerical Examples and Calculations
To further illustrate the concept of telescope coma aberration numericals, let’s consider a few numerical examples:
 Coma Aberration in a 100mm Telescope
 Focal length (f) = 1000mm
 Field of view (h) = 1 degree
 Zernike coefficient for coma (C₃₁) = (8/3)√(3/5) × (h/f)³ = 0.0025

Aberration level = 0.25% of image height (excellent performance)

Coma Aberration in a 200mm Telescope
 Focal length (f) = 2000mm
 Field of view (h) = 0.5 degree
 Zernike coefficient for coma (C₃₁) = (8/3)√(3/5) × (h/f)³ = 0.0006

Aberration level = 0.06% of image height (excellent performance)

Coma Aberration in a Fast Lens (f/2.8)
 Focal length (f) = 50mm
 Field of view (h) = 10 degrees
 Zernike coefficient for coma (C₃₁) = (8/3)√(3/5) × (h/f)³ = 0.0175
 Aberration level = 1.75% of image height (poor performance)
These examples demonstrate how the level of coma aberration can be calculated and quantified using the Zernike coefficient and the simple percentage scale. The results show that larger telescopes with longer focal lengths and smaller fields of view tend to have lower levels of coma aberration, while fast lenses with wider fields of view are more susceptible to this type of aberration.
Conclusion
Telescope coma aberration is a critical issue in optical systems that must be understood and quantified to design and optimize highperformance telescopes and lenses. By using the Zernike polynomial representation and the simple percentage scale, physicists and engineers can effectively measure and mitigate coma aberration, leading to improved image quality and enhanced astronomical observations.
References
 A Practical Guide to Lens Aberrations and the Lonely Speck Aberration Test: https://www.lonelyspeck.com/apracticalguidetolensaberrationsandthelonelyspeckaberrationtest/
 Aberration measurement and correction on a large field of view in adaptive optics: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8803008/
 Experimental investigation in nodal aberration theory (NAT): https://opg.optica.org/abstract.cfm?uri=oe30711150
 Optical Aberrations in Telescopes: https://www.astroshop.eu/knowledge/opticalaberrationsintelescopes/a,1032
 Zernike Polynomials and Optical Aberrations: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=7745
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