Astigmatism (optical systems)

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Astigmatism: The image of point G is at point BM for rays in plane M, and at point BS for rays in plane S.

An

optical system with astigmatism is one where rays that propagate in two perpendicular planes have different foci. If an optical system with astigmatism is used to form an image of a cross, the vertical and horizontal lines will be in sharp focus at two different distances. The term comes from the Greek α- (a-) meaning "without" and στίγμα (stigma), "a mark, spot, puncture".[1]

Forms of astigmatism

There are two distinct forms of astigmatism. The first is a third-order

aberration, which occurs for objects (or parts of objects) away from the optical axis. 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
. 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

lens
.

Third-order astigmatism

Page explaining and illustrating astigmatism[2]

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 particular planes. The first plane is the tangential plane. This is the plane which includes both the object point under consideration 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 plane used in the analysis is the sagittal plane. This is defined as the plane,

sagittal rays
.

In third-order astigmatism, the sagittal and transverse rays form

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 anastigmats.

Astigmatism in systems that are not rotationally symmetric

Visual astigmatism
Blur from astigmatic lens at different distances.

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. Some telescopes deliberately use non-spherical optics to overcome this phenomenon.[why?][3][failed verification]

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

Main article: Astigmatism

In

refraction error of the eye in which there is a difference in degree of refraction in different meridians.[4] 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 less than that of the perpendicular axis.

Astigmatism causes difficulties in seeing fine detail. Astigmatism can be often corrected by

lens that has different radii of curvature in different planes (a cylindrical lens), contact lenses, or refractive surgery.[5]

Astigmatism is quite common. Studies have shown that about one in three people suffers from it.

headaches.[10][11][12]

There are a number of tests that are used by

optometrists during eye examinations to determine the presence of astigmatism and to quantify the amount and axis of the astigmatism.[13] A Snellen chart or other eye chart may initially reveal reduced visual acuity. A keratometer may be used to measure the curvature of the steepest and flattest meridians in the cornea's front surface.[14] Corneal topography may also be used to obtain a more accurate representation of the cornea's shape.[15] An autorefractor or retinoscopy may provide an objective estimate of the eye's refractive error and the use of Jackson cross cylinders in a phoropter may be used to subjectively refine those measurements.[16][17][18] An alternative technique with the phoropter requires the use of a "clock dial" or "sunburst" chart to determine the astigmatic axis and power.[19][20]

Astigmatism may be corrected with

contact lenses, or refractive surgery.[21][22][23] Various considerations involving ocular health, refractive status, and lifestyle frequently determine whether one option may be better than another. In those with keratoconus, toric contact lenses often enable patients to achieve better visual acuities than eyeglasses. If the astigmatism is caused by a problem such as deformation of the eyeball due to a chalazion
, treating the underlying cause will resolve the astigmatism.

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

cube
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 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.[citation needed]

In 3D PALM/STORM, a type of optical super-resolution microscopy, a cylindrical lens can be introduced into the imaging system to create astigmatism, which allows measurement of the Z position of a diffraction-limited light source.[24]

Laser line levels use a cylindrical lens to spread a laser beam from a point into a line.

See also

References

  1. ^ Harper, Douglas (2001). "Online Etymology Dictionary". Retrieved 2007-12-29.
  2. ^ Frederic Eugene Wright (1911). The Methods of Petrographic-microscopic Research, Their Relative Accuracy and Range of Application. Carnegie institution of Washington.
  3. ^ Sacek, Vladimir (July 14, 2006). "Telescope astigmatism". Amateur Telescope Optics. Archived from the original on 19 September 2008. Retrieved Oct 16, 2008.
  4. ^ "Facts About Astigmatism | National Eye Institute". nei.nih.gov. Archived from the original on 2016-10-02. Retrieved 2019-05-06.
  5. ^ "Astigmatism Laser Eye Surgery". The Irish Times. Archived from the original on 2012-10-24. Retrieved 2011-08-29.
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  10. ^ Eyetopics.com
  11. ^ Medicinenet.com
  12. ^ Hipusa.com Archived May 1, 2006, at the Wayback Machine
  13. ^ Hipusa.com Archived April 26, 2006, at the Wayback Machine
  14. ^ Stlukeseye.com Archived March 23, 2006, at the Wayback Machine
  15. ^ Emedicine.com Archived February 18, 2006, at the Wayback Machine
  16. PMID 13900989
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  17. PMID 3808608
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  18. PMID 8320415
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  19. ^ Quantumoptical.com
  20. ^ Nova.edu
  21. ^ "Contact Lenses for Vision Correction". American Academy of Ophthalmology. 2018-09-01. Retrieved 2019-05-06.
  22. ^ "Eyeglasses for Vision Correction". American Academy of Ophthalmology. 2015-12-12. Retrieved 2019-05-06.
  23. ^ "LASIK eye surgery". Mayo Clinic. Retrieved 2019-05-06.
  24. PMID 18174397
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External links
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