Coma

Coma is an aberration which causes rays from an off-axis point of light in the object plane to create a trailing "comet-like" blur directed away from the optic axis (for positive coma). A lens with considerable coma may produce a sharp image in the center of the field, but become increasingly blurred toward the edges. For a single lens, coma can be partially corrected by bending the lens. More complete correction can be achieved by using a combination of lenses symmetric about a central stop. 像差是一种 aberration，它使物体平面中偏离光轴的点光源的光线形成一个向光轴方向延伸的‘彗星状’模糊区域（对于正像差）。一个存在显著像差的透镜可能在场内中心产生清晰的图像，但向边缘逐渐变得模糊。对于单透镜，可以通过弯曲透镜部分地校正像差。更彻底的校正可以通过使用关于中央光阑对称的透镜组合来实现。

Hecht, 2nd Ed in Sec 6.3 on aberrations gives a careful discussion of coma. For positive coma, rays from an off-axis object through larger radii of the lens contribute a larger oval of light further from the principal focus. Hecht 第二版第六章第三节中关于像差的讨论中，详细说明了彗差。对于正彗差，来自偏离光轴物体的光线通过透镜较大半径区域时，会形成更远离主焦点的更大光斑。

Hecht notes that a little more than half of the light in the image is in the roughly triangular region inside the large oval above, so the image fades as it expands away from the central focus. That encourages the picturing of this image as comet-like.

Coma is shape-dependent, so a lens shape can be found with zero coma for a given object distance. That lens would not be optimum for other object distances. 彗形像差与像的形状有关，因此可以找到一个在给定物距下具有零彗形像差的透镜。该透镜对于其他物距来说可能不是最优的。

For more versatile correction of coma, one can use a combination of two lenses, both of which are corrected for zero coma at infinite object distance. The appropriate separation of these two lenses can correct for coma at various object distances. Coma can also be corrected by an appropriately placed stop, but the placement and size of an optimum stop also depends upon the other aberrations.

Astigmatism

The kind of astigmatism commonly encountered as a vision defect is a result of different lens curvatures in different planes. If the behavior of light from a point source object is examined, then for a perfect lens it should form a focused bright spot on the opposite side of the lens. But if the focal length of the lens is different for different planes of incident light, there will be no point where all the rays from the object reach a sharp focus.

Oblique Astigmatism

The kind of astigmatism commonly encountered as a vision defect is a result of different lens curvatures in different planes. But even spherical lenses that are perfectly symmetrical exhibit a type of astigmatism for light which approaches the lens from a point off the optic axis. Oblique astigmatism is an aberration of off-axis rays that causes radial and tangential lines in the object plane to focus sharply at different distances in the image space.

This visualization of oblique astigmatism follows the treatment of Jenkins and White in their text "Fundamentals of Optics", 4th. Ed., Sec. 9.9 . Similar treatments can be found in Pedrotti & Pedrotti, Sec 5-5 and Hecht, 2nd Ed Sec 6.3 . Oblique astigmatism arises not from any asymmetry of the lens, but an asymmetry in the nature of the optical paths followed by rays in the tangential and sagittal planes. For the sagittal plane, the extreme rays are symmetric with respect to the central ray. For the tangential plane however, the pathlengths and angles of incidence on the lens surface are different for the top and bottom rays. (Some texts use the term "meridional plane" instead of tangential plane and "radial plane" instead of sagittal plane.)

The focal distance shortens relative to that for rays along the axis by an amount approximately proportional to the square of the displacement from the axis, forming a paraboloid of revolution as the object is rotated around the axis. 相对于轴线上光线的焦距，随着物体绕轴旋转，由于与轴的位移平方成比例地缩短，形成一个旋转抛物面。

Oblique Astigmatism Examples

Meyer-Arendt's text does a careful job of trying to visualize the effects of oblique astigmatism in Ch 1.6. If an object consists of a short line segment perpendicular to the radial line from the optic axis (like a segment tangential to a circle centered at the optic axis in the object plane), then that line segment will image sharply in the tangential image plane. If an object consists of a short segment in the sagittal plane, like a short radial segment in the object plane, it will image sharply in the sagittal image plane.

This illustration seeks to show that radial line segments will focus sharply in the sagittal image plane while tangential line segments will focus sharply in the tangential image plane. The result is that different patterns of distortion from oblique astigmatism will appear depending upon whether the plane of projection of the image is closer to the tangential or the sagittal image plane. 这张图旨在说明，径向线段会在视向像平面中聚焦得非常清晰，而切向线段则会在切向像平面中聚焦得非常清晰。结果是，根据图像平面是更接近切向像平面还是视向像平面，不同类型的斜向散光畸变模式会显现出来。

This illustration follows Meyer-Arendt's approach to showing the different types of distortion of the image by oblique astigmatism when viewed in the tangential and sagittal focus planes.

Another approach to visualizing the effects of oblique astigmatism on the image is employed by Moller, Chapter 13. For an object like the spoked wheel shown, the circumference is imaged sharply in the tangential focus plane and the radial features are imaged sharply in the sagittal focus plane. At some point in between, each image point can be represented by a "circle of least confusion" and the overall best focus for the object is obtained at that point.