Astigmatism

# Astigmatism

Discussion

Encyclopedia
An optical system with astigmatism is one where rays
Ray (optics)
In optics, a ray is an idealized narrow beam of light. Rays are used to model the propagation of light through an optical system, by dividing the real light field up into discrete rays that can be computationally propagated through the system by the techniques of ray tracing. This allows even very...

that propagate in two perpendicular planes
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

have different foci
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

. If an optical system with astigmatism is used to form an image of a cross
Cross
A cross is a geometrical figure consisting of two lines or bars perpendicular to each other, dividing one or two of the lines in half. The lines usually run vertically and horizontally; if they run obliquely, the design is technically termed a saltire, although the arms of a saltire need not meet...

, the vertical and horizontal lines will be in sharp focus at two different distances. The term comes from the Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...

α- (a-) meaning "without" and στίγμα (stigma), "a mark, spot, puncture".

## Forms of astigmatism

There are two distinct forms of astigmatism. The first is a third-order aberration
Aberration in optical systems
Aberrations are departures of the performance of an optical system from the predictions of paraxial optics. Aberration leads to blurring of the image produced by an image-forming optical system. It occurs when light from one point of an object after transmission through the system does not converge...

, which occurs for objects (or parts of objects) away from the optical axis
Optical axis
An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens or microscope.The optical axis is an imaginary line that defines the path along which light propagates through the system...

. This form of aberration occurs even when the optical system is perfectly symmetrical. This is often referred to as a "monochromatic aberration", because it occurs even for light of a single wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

. This terminology may be misleading, however, as the amount of aberration can vary strongly with wavelength in an optical system.

The second form of astigmatism occurs when the optical system is not symmetric about the optical axis. This may be by design (as in the case of a cylindrical
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...

lens), or due to manufacturing error in the surfaces of the components or misalignment of the components. In this case, astigmatism is observed even for rays from on-axis object points. This form of astigmatism is extremely important in vision science and eye care, since the human eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

often exhibits this aberration due to imperfections in the shape of the cornea
Cornea
The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Together with the lens, the cornea refracts light, with the cornea accounting for approximately two-thirds of the eye's total optical power. In humans, the refractive power of the cornea is...

or the lens
Lens (anatomy)
The crystalline lens is a transparent, biconvex structure in the eye that, along with the cornea, helps to refract light to be focused on the retina. The lens, by changing shape, functions to change the focal distance of the eye so that it can focus on objects at various distances, thus allowing a...

.

### Third-order astigmatism

In the analysis of this form of astigmatism, it is most common to consider rays from a given point on the object, which propagate in two special planes. The first plane is the tangential plane. This is the plane which includes both the object point being considered and the axis of symmetry. Rays that propagate in this plane are called tangential rays. Planes that include the optical axis are meridional planes. It is common to simplify problems in radially-symmetric optical systems by choosing object points in the vertical ("y") plane only. This plane is then sometimes referred to as the meridional plane.

The second special plane is the sagittal plane. This is defined as the plane, orthogonal to the tangential plane, which contains the object point being considered and intersects the optical axis at the entrance pupil
Entrance pupil
In an optical system, the entrance pupil is the optical image of the physical aperture stop, as 'seen' through the front of the lens system. The corresponding image of the aperture as seen through the back of the lens system is called the exit pupil...

of the optical system. This plane contains the chief ray, but does not contain the optic axis. It is therefore a skew plane, in other words not a meridional plane. Rays propagating in this plane are called sagittal rays.

In third-order astigmatism, the sagittal and transverse rays form foci
Focus (optics)
In geometrical optics, a focus, also called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle. This non-ideal focusing may be caused by...

at different distances along the optic axis. These foci are called the sagittal focus and the transverse focus, respectively. In the presence of astigmatism, an off-axis point on the object is not sharply imaged by the optical system. Instead, sharp lines are formed at the sagittal and transverse foci. The image at the transverse focus is a short line, oriented in the direction of the sagittal plane; images of circles centered on the optic axis, or lines tangential to such circles, will be sharp in this plane. The image at the sagittal focus is a short line, oriented in the tangential direction; images of spokes radiating from the center are sharp at this focus. In between these two foci, a round but "blurry" image is formed. This is called the medial focus or circle of least confusion. This plane often represents the best compromise image location in a system with astigmatism.

The amount of aberration due to astigmatism is proportional to the square of the angle between the rays from the object and the optical axis of the system. With care, an optical system can be designed to reduce or eliminate astigmatism. Such systems are called anastigmat
Anastigmat
An Anastigmat or anastigmatic lens is a photographic lens completely corrected for spherical aberration, coma, and astigmatism. Early lenses often included the word Anastigmat in their name to advertise this new feature...

s.

### Astigmatism in systems that are not rotationally symmetric

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 telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

.

In the analysis of these systems, it is common to consider tangential rays (as defined above), and rays in a meridional plane (a plane containing the optic axis) perpendicular to the tangential plane. This plane is called either the sagittal meridional plane or, confusingly, just the sagittal plane.

#### Ophthalmic astigmatism

In optometry
Optometry
Optometry is a health care profession concerned with eyes and related structures, as well as vision, visual systems, and vision information processing in humans. Optometrists, or Doctors of Optometry, are state licensed medical professionals trained to prescribe and fit lenses to improve vision,...

and ophthalmology
Ophthalmology
Ophthalmology is the branch of medicine that deals with the anatomy, physiology and diseases of the eye. An ophthalmologist is a specialist in medical and surgical eye problems...

, the vertical and horizontal planes are identified as tangential and sagittal meridians, respectively. Ophthalmic astigmatism is a refraction error of the eye
Human eye
The human eye is an organ which reacts to light for several purposes. As a conscious sense organ, the eye allows vision. Rod and cone cells in the retina allow conscious light perception and vision including color differentiation and the perception of depth...

in which there is a difference in degree of refraction
Refraction
Refraction is the change in direction of a wave due to a change in its speed. It is essentially a surface phenomenon . The phenomenon is mainly in governance to the law of conservation of energy. The proper explanation would be that due to change of medium, the phase velocity of the wave is changed...

in different meridians. It is typically characterized by an aspherical, non-figure of revolution cornea
Cornea
The cornea is the transparent front part of the eye that covers the iris, pupil, and anterior chamber. Together with the lens, the cornea refracts light, with the cornea accounting for approximately two-thirds of the eye's total optical power. In humans, the refractive power of the cornea is...

in which the corneal profile slope
Slope
In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

and refractive power in one meridian is less than that of the perpendicular axis.

Astigmatism causes difficulties in seeing fine detail. Astigmatism can be often corrected by glasses
Glasses
Glasses, also known as eyeglasses , spectacles or simply specs , are frames bearing lenses worn in front of the eyes. They are normally used for vision correction or eye protection. Safety glasses are a kind of eye protection against flying debris or against visible and near visible light or...

with a lens
Lens (optics)
A lens is an optical device with perfect or approximate axial symmetry which transmits and refracts light, converging or diverging the beam. A simple lens consists of a single optical element...

that has different radii of curvature
Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

in different planes (a cylindrical lens), contact lens
Contact lens
A contact lens, or simply contact, is a lens placed on the eye. They are considered medical devices and can be worn to correct vision, for cosmetic or therapeutic reasons. In 2004, it was estimated that 125 million people use contact lenses worldwide, including 28 to 38 million in the United...

es, or refractive surgery
Refractive surgery
Refractive eye surgery is any eye surgery used to improve the refractive state of the eye and decrease or eliminate dependency on glasses or contact lenses. This can include various methods of surgical remodeling of the cornea or cataract surgery. The most common methods today use excimer lasers to...

.

Astigmatism is quite common. Studies have shown that about one in three people suffers 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
Asthenopia
Asthenopia or eye strain is an ophthalmological condition that manifests itself through nonspecific symptoms such as fatigue, pain in or around the eyes, blurred vision, headache and occasional double vision...

, fatigue
Fatigue (physical)
Fatigue is a state of awareness describing a range of afflictions, usually associated with physical and/or mental weakness, though varying from a general state of lethargy to a specific work-induced burning sensation within one's muscles...

There are a number of tests used by ophthalmologists and optometrists during eye examination
Eye examination
An eye examination is a battery of tests performed by an ophthalmologist, optometrist, or orthoptist assessing vision and ability to focus on and discern objects, as well as other tests and examinations pertaining to the eyes....

s to determine the presence of astigmatism and to quantify the amount and axis of the astigmatism. A Snellen chart
Snellen chart
A Snellen chart is an eye chart used by eye care professionals and others to measure visual acuity. Snellen charts are named after the Dutch ophthalmologist Herman Snellen who developed the chart during 1862...

or other eye chart
Eye chart
An eye chart is a chart used to measure visual acuity. Types of eye charts include the logMAR chart, Snellen chart, Landolt C, and the Lea test.-Procedure:Charts usually display several rows of optotypes , each row in a different size...

may initially reveal reduced visual acuity
Visual acuity
Visual acuity is acuteness or clearness of vision, which is dependent on the sharpness of the retinal focus within the eye and the sensitivity of the interpretative faculty of the brain....

. A keratometer
Keratometer
A keratometer, also known as a ophthalmometer, is a diagnostic instrument for measuring the curvature of the anterior surface of the cornea, particularly for assessing the extent and axis of astigmatism...

may be used to measure the curvature of the steepest and flattest meridians in the cornea's front surface. Corneal topography
Corneal topography
Corneal topography, also known as photokeratoscopy or videokeratography, is a non-invasive medical imaging technique for mapping the surface curvature of the cornea, the outer structure of the eye...

may also be used to obtain a more accurate representation of the cornea's shape. An autorefractor or retinoscopy
Retinoscopy
Retinoscopy is a technique to obtain an objective measurement of the refractive error of a patient's eyes. The examiner uses a retinoscope to shine light into the patient's eye and observes the reflection off the patient's retina...

may provide an objective estimate of the eye's refractive error and the use of Jackson cross cylinders in a phoropter
Phoropter
A phoropter is an instrument commonly used by eye care professionals during an eye examination, containing different lenses used for refraction of the eye during sight testing, to measure an individual's refractive error and determine his or her eyeglass prescription.Typically, the patient sits...

may be used to subjectively refine those measurements. An alternative technique with the phoropter requires the use of a "clock dial" or "sunburst" chart to determine the astigmatic axis and power.

#### Astigmatism due to misaligned or malformed lenses and mirrors

Grinding and polishing of precision optical parts, either by hand or machine, typically employs significant downward pressure, which in turn creates significant frictional side pressures during polishing strokes that can combine to locally flex and distort the parts. These distortions generally do not possess figure-of-revolution symmetry and are thus astigmatic, and slowly become permanently polished into the surface if the problems causing the distortion are not corrected. Astigmatic, distorted surfaces potentially introduce serious degradations in optical system performance.

Surface distortion due to grinding or polishing increases with the aspect ratio
Aspect ratio
The aspect ratio of a shape is the ratio of its longer dimension to its shorter dimension. It may be applied to two characteristic dimensions of a three-dimensional shape, such as the ratio of the longest and shortest axis, or for symmetrical objects that are described by just two measurements,...

of the part (diameter to thickness ratio). To a first order, glass strength increases as the cube
Cube (arithmetic)
In arithmetic and algebra, the cube of a number n is its third power — the result of the number multiplying by itself three times:...

of the thickness. Thick lenses at 4:1 to 6:1 aspect ratios will flex much less than high aspect ratio parts, such as optical windows, which can have aspect ratios of 15:1 or higher. The combination of surface or wavefront error precision requirements and part aspect ratio drives the degree of back support uniformity required, especially during the higher down pressures and side forces during polishing. Optical working typically involves a degree of randomness that helps greatly in preserving figure-of-revolution surfaces, provided the part is not flexing during the grind/polish process.

#### Deliberate astigmatism in optical systems

Compact disc
Compact Disc
The Compact Disc is an optical disc used to store digital data. It was originally developed to store and playback sound recordings exclusively, but later expanded to encompass data storage , write-once audio and data storage , rewritable media , Video Compact Discs , Super Video Compact Discs ,...

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 which 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 best focus, without being fooled by oblong pits or other features on the disc surface.

Some telescope
Telescope
A telescope is an instrument that aids in the observation of remote objects by collecting electromagnetic radiation . The first known practical telescopes were invented in the Netherlands at the beginning of the 1600s , using glass lenses...

s use deliberately astigmatic optics.