Astigmatism

Astigmatism is a phenomenon observed in optics when an optical system presents different foci for rays propagating in two perpendicular planes. When an optical system demonstrating astigmatism is employed to form an image of a cross, the vertical and horizontal lines appear in sharp focus at two distinct distances.

Astigmatism can manifest in two distinct forms. The first is third-order aberration, which is observed for objects or parts of objects located away from the optical axis. Despite the perfect symmetry of the optical system, this form of aberration still occurs. Often called "monochromatic aberration", this term can be misleading as the intensity of aberration can strongly vary with wavelength within the optical system.

The second form of astigmatism emerges when the optical system lacks symmetry around the optical axis. This could be by design, as seen in a cylindrical lens, or due to manufacturing errors in the surfaces of the components or misalignment of the components. In this scenario, astigmatism is observed even for rays from on-axis object points. This form of astigmatism is of significant importance in ophthalmology, given that the human eye often exhibits this aberration due to imperfections in the shape of the cornea or the lens.

Types of Astigmatism

Third-order Astigmatism

SVC Insight - Correcting for Astigmatism
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In the analysis of third-order astigmatism, it is common to examine rays from a particular point on the object that propagate in two special planes. The first plane, the tangential plane, includes both the object point under consideration and the axis of symmetry. Rays that propagate in this plane are called tangential rays. The second plane is the sagittal plane, defined as the plane orthogonal to the tangential plane, containing the object point under consideration and intersecting the optical axis at the entrance pupil of the optical system. Rays propagating in this plane are called sagittal rays.

In third-order astigmatism, the sagittal and transverse rays form foci at different distances along the optic axis, known as the sagittal focus and the transverse focus. An off-axis point on the object is not sharply imaged by the optical system under astigmatism, instead, sharp lines are formed at the sagittal and transverse foci. In between these two foci, a round but "blurry" image is formed, called the medial focus or circle of least confusion. This plane often represents the best compromise image location in a system with astigmatism.

The degree of aberration resulting from astigmatism is proportional to the square of the angle between the rays from the object and the optical axis of the system. With meticulous design, an optical system can be designed to reduce or eliminate astigmatism, known as anastigmats.

Astigmatism in Non-Rotationally Symmetric Systems

If an optical system is not axisymmetric, either due to an error in the shape of the optical surfaces or due to misalignment of the components, astigmatism can occur even for on-axis object points. This effect is often used deliberately in complex optical systems, especially certain types of telescopes.

Ophthalmic Astigmatism

In ophthalmology, the vertical and horizontal planes are referred to as tangential and sagittal meridians, respectively. Ophthalmic astigmatism is a refraction error of the eye where there is a difference in the degree of refraction in different meridians. It is typically characterized by an aspherical, non-figure of revolution cornea in which the corneal profile slope and refractive power in one meridian is greater than that of the perpendicular axis.

Astigmatism can cause difficulties in seeing fine detail. For instance, vertical lines and objects such as walls may appear to the patient to be leaning over, akin to the Tower of Pisa. Astigmatism can be corrected by glasses with a lens that has different radii of curvature in different planes (a cylindrical lens), contact lenses, or refractive surgery.

Astigmatism is quite prevalent. Studies have shown that about one in three people suffer from it. The prevalence of astigmatism increases with age. Although a person may not notice mild astigmatism, higher amounts of astigmatism may cause blurry vision, squinting, asthenopia, fatigue, or headaches.

During eye examinations, ophthalmologists and optometrists use a variety of tests to determine the presence of astigmatism and to quantify the amount and axis of the astigmatism. A Snellen chart or other eye chart may initially reveal reduced visual acuity. Instruments such as a keratometer may be used to measure the curvature of the steepest and flattest meridians in the cornea's front surface. An autorefractor or retinoscopy may provide an objective estimate of the eye's refractive error.

Astigmatism due to Misaligned or Malformed Lenses and Mirrors

Astigmatism can also result from misaligned or malformed lenses and mirrors. This type of astigmatism can significantly affect the quality of the image produced by the optical system. It can often be corrected by realigning the optical elements or replacing the defective components.

Deliberate Astigmatism in Optical Systems

Deliberate astigmatism is sometimes used in optical systems for specific purposes. For instance, Compact disc players use an astigmatic lens for focusing. When one axis is more in focus than the other, dot-like features on the disc project to oval shapes. The orientation of the oval indicates which axis is more in focus, and thus the direction the lens needs to move. A square arrangement of only four sensors can observe this bias and use it to bring the read lens to the best focus, without being fooled by oblong pits or other features on the disc surface.

Some telescopes also use deliberately astigmatic optics to achieve specific imaging objectives.

Astigmatism Resources
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