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dioptre
Unit ofoptical power
Symboldpt, D
Conversions
1 dpt in ...... is equal to ...
   SI units   1 m−1
Illustration of the relationship between optical power in dioptres and focal length in metres.

A dioptre (British spelling) or diopter (American spelling), symbol dpt or D, is a unit of measurement with dimension of reciprocal length, equivalent to one reciprocal metre, 1 dpt = 1 m−1. It is normally used to express the optical power of a lens or curved mirror, which is a physical quantity equal to the reciprocal of the focal length, expressed in metres. For example, a 3-dioptre lens brings parallel rays of light to focus at 1⁄3 metre. A flat window has an optical power of zero dioptres, as it does not cause light to converge or diverge. Dioptres are also sometimes used for other reciprocals of distance, particularly radii of curvature and the vergence of optical beams.

The main benefit of using optical power rather than focal length is that the thin lens formula has the object distance, image distance, and focal length all as reciprocals. Additionally, when relatively thin lenses are placed close together their powers approximately add. Thus, a thin 2.0-dioptre lens placed close to a thin 0.5-dioptre lens yields almost the same focal length as a single 2.5-dioptre lens.

The idea of numbering lenses based on the reciprocal of their focal length in metres was first suggested by Albrecht Nagel in 1866.[1][2] The term dioptre was proposed by French ophthalmologist Ferdinand Monoyer in 1872, based on earlier use of the term dioptrice by Johannes Kepler.[3][4][5]

Name, symbol

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Though the dioptre is based on the SI-metric system, it has not been included in the standard, so that there is no international name or symbol for this unit of measurement – within the international system of units, this unit for optical power would need to be specified explicitly as the inverse metre (m−1). However most languages have borrowed the original name and some national standardization bodies like DIN specify a unit name (dioptrie, dioptria, etc.). In vision care the symbol D is frequently used.

In vision correction

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The fact that optical powers are approximately additive enables an eye care professional to prescribe corrective lenses as a simple correction to the eye's optical power, rather than doing a detailed analysis of the entire optical system (the eye and the lens). Optical power can also be used to adjust a basic prescription for reading. Thus an eye care professional, having determined that a myopic (nearsighted) person requires a basic correction of, say, −2 dioptres to restore normal distance vision, might then make a further prescription of 'add 1' for reading, to make up for lack of accommodation (ability to alter focus). This is the same as saying that −1 dioptre lenses are prescribed for reading.

In humans, the total optical power of the relaxed eye is approximately 60 dioptres.[6][7] The cornea accounts for approximately two-thirds of this refractive power (about 40 dioptres) and the crystalline lens contributes the remaining one-third (about 20 dioptres).[6] In focusing, the ciliary muscle contracts to reduce the tension or stress transferred to the lens by the suspensory ligaments. This results in increased convexity of the lens which in turn increases the optical power of the eye. The amplitude of accommodation is about 11 to 16 dioptres at age 15, decreasing to about 10 dioptres at age 25, and to around 1 dioptre above age 60.

Convex lenses have positive dioptric value and are generally used to correct hyperopia (farsightedness) or to allow people with presbyopia (the limited accommodation of advancing age) to read at close range. Over the counter reading glasses are rated at +1.00 to +4.00 dioptres. Concave lenses have negative dioptric value and generally correct myopia (nearsightedness). Typical glasses for mild myopia have a power of −0.50 to −3.00 dioptres. Optometrists usually measure refractive error using lenses graded in steps of 0.25 dioptres.

Curvature

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The dioptre can also be used as a measurement of curvature equal to the reciprocal of the radius measured in metres. For example, a circle with a radius of 1/2 metre has a curvature of 2 dioptres. If the curvature of a surface of a lens is C and the index of refraction is n, the optical power is φ = (n − 1)C. If both surfaces of the lens are curved, consider their curvatures as positive toward the lens and add them. This gives approximately the right result, as long as the thickness of the lens is much less than the radius of curvature of one of the surfaces. For a mirror the optical power is φ = 2C.

Relation to magnifying power

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Further information: Magnification and Magnifying glass § Magnification

The magnifying power V of a simple magnifying glass is related to its optical power φ by

.

This is approximately the magnification observed when a person with normal vision holds the magnifying glass close to his or her eye.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Dioptre

View on Grokipedia
from Grokipedia
A dioptre (British English; symbol: D or dpt) is a in that quantifies the power of a lens or , defined as the reciprocal of the in metres. For example, a lens with a of 0.5 m has an of 2 dioptres, while one with a of 2 m has 0.5 dioptres. Positive values indicate converging lenses that focus light, whereas negative values denote diverging lenses that spread it out. The dioptre was proposed in 1872 by French ophthalmologist Ferdinand Monoyer as a standardized unit for measuring refractive errors in the eye, building on earlier advancements like the ophthalmoscope (invented in 1850) and the Snellen acuity chart (introduced in 1862). Prior to this, spectacle lenses in the late 16th century were graded roughly by focal power, but lacked a uniform metric like the dioptre. Monoyer's system enabled precise quantification, equivalent to 1/f (where f is focal length in metres), or alternatively 100/f in centimetres or 40/f in inches for practical conversions. In modern applications, dioptres are essential in and for prescribing corrective , where lens powers are typically specified to the nearest quarter dioptre to address conditions like (nearsightedness, negative dioptres) or hyperopia (, positive dioptres). They also describe the eye's accommodative amplitude—the range over which the lens can adjust focus—and vergence in optical systems. Beyond vision correction, the unit applies to any refractive element, underscoring its foundational role in and physics.

Fundamentals

Definition

The dioptre (symbol: ) is a used to quantify the of a lens or . in this context refers to the degree to which the element bends rays, either converging them to a focus or diverging them. It is defined as the reciprocal of the in meters, expressed by the formula P=1/fP = 1/f, where PP is the in dioptres and ff is the in meters; thus, a lens with a of 1 m has an of 1 . Positive values indicate converging elements, such as convex lenses that focus parallel rays to a point, while negative values denote diverging elements, like concave lenses that spread rays apart. Unlike length units such as the or , the dioptre specifically measures the inverse of , with 1 D equivalent to a 1-meter focal length, providing a standardized way to compare the bending strength across optical components. For instance, a lens with +2 D power converges parallel incident rays to a focal point 0.5 m away. This unit is particularly useful in fields like vision correction, where lens prescriptions are specified in dioptres.

Etymology and History

The term "dioptre" originates from the ancient Greek word dioptra (διόπτρα), referring to an optical sighting instrument used in to measure angles, altitudes, and distances. This device, possibly invented by the astronomer around 150 BC, was later detailed by in his first-century AD treatise Dioptra, where it is described as a precision tool combining sighting mechanisms with water levels for accurate geodetic work. The instrument's name, meaning "to see through," reflected its function in aligning sights over long distances, marking an early application of optical principles in measurement. The modern dioptre as a unit of optical power emerged in the 19th century amid advances in ophthalmology and lens manufacturing. Prior to its adoption, lens strengths were empirically denoted by focal length, often in inches or arbitrary scales, complicating precise prescriptions and standardization. French ophthalmologist Ferdinand Monoyer formalized the dioptre in 1872, defining it as the reciprocal of the focal length in meters to quantify lens refractive power systematically. This innovation built on earlier conceptual work, including Johannes Kepler's 17th-century use of "dioptrice" for optical computations, reviving the ancient term for practical clinical use. By around 1875, the dioptre gained traction in , enabling standardized trial lens sets and refraction techniques that replaced inconsistent notations. Integrated into the , it aligned with the centimeter-gram-second (CGS) framework prevalent in 19th-century science before the (SI) was established in 1960, where the dioptre remains a derived unit (m⁻¹) accepted for optical measurements. This evolution facilitated global consistency in vision correction, though early adoption varied by region until broader metric standardization.

Optical Properties

Relation to Focal Length

The optical power PP of a lens, measured in dioptres (D), is defined as the reciprocal of its focal length ff in metres: P=1fP = \frac{1}{f}. This relationship directly links the converging or diverging ability of the lens to its geometry and material properties, with the focal length representing the distance from the lens to the point where parallel rays converge or appear to diverge. Rearranging gives f=1Pf = \frac{1}{P}, allowing focal length to be computed straightforwardly from the power. For a spherical mirror, the is P=2RP = \frac{2}{R}, where RR is the in metres, with positive values for concave (converging) mirrors and negative for convex (diverging) ones, following the Cartesian . A key aspect of this relation is the , which follows the Cartesian sign rule in paraxial : positive focal lengths apply to converging lenses that form real images for distant objects, while negative focal lengths apply to diverging lenses that form virtual images. For instance, a converging lens with f=+0.5f = +0.5 m has P=+2P = +2 D, focusing parallel rays to a real point 50 cm away on the opposite side. Conversely, a diverging lens with f=−0.5f = -0.5 m has P=−2P = -2 D, causing parallel rays to appear to originate from a virtual point 50 cm on the same side. For systems of multiple thin lenses, the thin lens approximation simplifies calculations by treating lenses as infinitesimally thin and neglecting their thickness. Under this , the total power PtotalP_\text{total} of two thin lenses with powers P1P_1 and P2P_2, separated by a dd in metres, is given by: Ptotal=P1+P2−d⋅P1⋅P2P_\text{total} = P_1 + P_2 - d \cdot P_1 \cdot P_2 When d=0d = 0 (lenses in contact), this reduces to Ptotal=P1+P2P_\text{total} = P_1 + P_2, showing that powers add algebraically for touching thin lenses. The equivalent is then ftotal=1Ptotalf_\text{total} = \frac{1}{P_\text{total}}, maintaining the sign convention for the system's overall behaviour. The dioptre adheres to the (SI), where 1 D = 1 m−1^{-1}, emphasizing metres for consistency in optical calculations. In some practical contexts, the power is calculated as P=100/fP = 100 / f where ff is the in centimetres, but the SI unit remains 1 D = 1 m−1^{-1} to avoid scaling errors in derivations and measurements. A practical example is a lens with P=−3P = -3 D, commonly prescribed for moderate myopia (nearsightedness) correction, where the focal length is f=1−3=−0.333f = \frac{1}{-3} = -0.333 m ≈ -33 cm. This negative value indicates a diverging lens that shifts the focus of distant objects to a virtual image at the patient's far point, typically around 33 cm, enabling clear vision when combined with the eye's optics. In contrast, a +3 D lens for hyperopia (farsightedness) has f=+33f = +33 cm, using a converging lens to bring nearby objects into focus on the retina.

Lens Curvature

The optical power PP of a in air, measured in dioptres, is directly related to the curvatures of its surfaces through the lensmaker's equation: P=(n−1)(1R1−1R2),P = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right), where nn is the of the lens material, and R1R_1 and R2R_2 are the radii of of the first and second surfaces, respectively, in meters. This equation derives from the at spherical interfaces, assuming the lens thickness is negligible compared to the radii of . The sign convention for the radii follows the Cartesian rule: RR is positive if the center of curvature lies to the right of the surface (for light incident from the left), which corresponds to a convex surface facing the incident , and negative for a concave surface. For a typical biconvex lens, R1R_1 is positive and R2R_2 is negative, yielding a positive power for converging lenses. As an illustrative example, consider a symmetric biconvex lens made of with n=1.5n = 1.5 and both surfaces having a ∣R∣=0.2|R| = 0.2 m (R1=+0.2R_1 = +0.2 m, R2=−0.2R_2 = -0.2 m). Substituting into the equation gives P=(1.5−1)(10.2−1−0.2)=0.5×(5+5)=5 D.P = (1.5 - 1) \left( \frac{1}{0.2} - \frac{1}{-0.2} \right) = 0.5 \times (5 + 5) = 5~\text{D}. This results in a of 0.2 m, demonstrating how surface curvature determines the lens's converging strength. Beyond curvature, the nn significantly influences power, as higher nn amplifies the effect of a given (e.g., crown glass at n≈1.52n \approx 1.52 versus at n≈1.65n \approx 1.65). For thicker lenses, deviations arise because the thin-lens neglects the axial separation between surfaces, requiring more complex formulas that incorporate thickness dd to accurately compute effective power.

Applications

Vision Correction

In optometry, the dioptre serves as the primary unit for specifying lens power in eyeglass and contact lens prescriptions to correct common refractive errors. For myopia, or nearsightedness, a negative dioptre value indicates the need for diverging (concave) lenses, while positive dioptres prescribe converging (convex) lenses for hyperopia, or farsightedness. Astigmatism is addressed through cylindrical lens power, also measured in dioptres, combined with an axis angle to orient the correction for irregular corneal curvature. Dioptres quantify the degree of by measuring the additional required to focus light precisely on the , with representing perfect vision at 0 dioptres. For instance, a -4 dioptre prescription for means the eye focuses distant objects 25 cm in front of the , necessitating a diverging lens of equal power to shift the focus back to the retinal plane. This system allows precise matching of lens power to the eye's ametropia, ensuring clear vision for both distance and near tasks. Refractive errors are typically measured using objective techniques like , where an examiner observes the reflex from the 's to neutralize the error with trial lenses in dioptres, or via a phoropter, in which the compares lens options to refine the prescription. The least distance of distinct vision for young adults with normal accommodation is standardized at 25 cm, equivalent to 4 dioptres of accommodative amplitude, beyond which strain occurs without correction. The dioptre was proposed by Ferdinand Monoyer in 1872, discussed at the 4th International of Ophthalmology in that year, and supported at the 5th in New York in 1876, with widespread use solidified by early 20th-century metric conventions. In modern practice, prescriptions exceeding ±10 dioptres often require high-index materials to minimize lens thickness, though they can introduce peripheral distortions and aberrations that affect visual quality.

Magnification

In the context of simple magnifiers, such as loupes or handheld lenses, the dioptre measure of lens power directly influences the angular , which quantifies the increase in the apparent angular size of an object as perceived by the eye. MM is defined as the ratio of the angle subtended by the through the lens to the angle subtended by the object when viewed unaided at the least distance of distinct vision, typically 25 cm. For a simple magnifier forming a at this (accommodated eye), the formula is M=1+DfM = 1 + \frac{D}{f}, where D=0.25D = 0.25 m is the distance and ff is the in meters. Since the lens power PP in dioptres is P=1fP = \frac{1}{f}, this simplifies to M=1+0.25PM = 1 + 0.25 P. For viewing with a relaxed eye (image at infinity), the angular magnification reduces to M≈0.25PM \approx 0.25 P, as the eye does not accommodate. In practical applications like low-vision aids or precision loupes, magnification is often approximated as M=P4M = \frac{P}{4} for the relaxed case, reflecting the 25 cm standard . For example, a +10 D lens (f=0.1f = 0.1 m) yields M=1+0.25×10=3.5M = 1 + 0.25 \times 10 = 3.5 for the accommodated eye when the object is at approximately 25 cm, providing roughly 3.5× compared to unaided viewing at the same distance; this is sometimes rounded to about 4× in adjusted viewing contexts. This dioptre-based approach emphasizes angular magnification, which enhances the perceived size through increased rather than linear (transverse) magnification, where the image height scales directly with object distance ratios. In visual instruments like loupes, angular magnification is key, as it allows closer object placement without proportionally enlarging the physical image size, distinguishing it from linear magnification used in imaging systems like microscopes. Compared to the unaided eye viewing an object at (effective 1× angular size for distant objects), a simple magnifier dramatically increases the angular subtense, enabling detailed inspection of small features. Higher dioptre values, while boosting , introduce limitations such as increased optical aberrations, particularly spherical and chromatic, which degrade image quality by blurring edges and introducing color fringing. For instance, single-element magnifiers above +20 D often require multi-lens designs or coatings to mitigate these effects, as aberrations scale with power and size.

References

  1. http://physics.bu.edu/~duffy/sc528_notes11/diopters.html
  2. https://webeye.ophth.uiowa.edu/eyeforum/video/Refraction/Intro-Optics-Refract-Errors/index.htm
  3. https://pmc.ncbi.nlm.nih.gov/articles/PMC7532918/
  4. https://www.rp-photonics.com/dioptric_power.html
  5. https://www.azooptics.com/Article.aspx?ArticleID=77
  6. https://www.britannica.com/topic/Dioptra
  7. https://pmc.ncbi.nlm.nih.gov/articles/PMC8225652/
  8. https://www.merriam-webster.com/dictionary/diopter
  9. https://eyeantiques.com/shop/museum/eye-exam-equipment-museum/early-trial-lens-set-1874/
  10. https://onlinelibrary.wiley.com/doi/10.1111/opo.12211
  11. https://www.2020mag.com/ce/ophthalmic-lenses---then-and-n
  12. http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/foclen.html
  13. http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/mirpow.html
  14. https://www.cis.rit.edu/class/simg232/lab3.pdf
  15. https://shileyeye.ucsd.edu/eye-conditions/retinal-diseases/myopia
  16. http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/lenmak.html
  17. https://www.ncbi.nlm.nih.gov/books/NBK594278/
  18. https://openstax.org/books/university-physics-volume-3/pages/2-5-the-law-of-refraction
  19. https://www.translatorscafe.com/unit-converter/en-US/calculator/lensmaker-equation/
  20. https://flexbooks.ck12.org/cbook/cbse-physics-class-10/section/1.11/primary/lesson/power-of-a-lens/
  21. https://www.chop.edu/conditions-diseases/eyeglasses-and-contact-lenses
  22. https://www.aoa.org/healthy-eyes/caring-for-your-eyes/corneal-modifications
  23. https://webeye.ophth.uiowa.edu/eyeforum/tutorials/optics-and-refraction-medical-students/index.htm
  24. https://www.aao.org/young-ophthalmologists/yo-info/article/retinoscopy-101
  25. https://eyewiki.org/Retinoscopy
  26. https://www.ncbi.nlm.nih.gov/books/NBK597373/
  27. https://www.bantonframeworks.co.uk/blogs/eye-care/high-index-lenses
  28. https://www.physicsforums.com/threads/eyeglasses-with-smallest-chromatic-abberation.609223/
  29. https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.08%3A_The_Simple_Magnifier
  30. https://webeye.ophth.uiowa.edu/eyeforum/video/Refraction/pdfs/Optics-Review.pdf
  31. https://hadley.edu/sites/default/files/2020-10/Magnifiers_LowVisionMagnificationDevice_HO_ACC.pdf
  32. https://www.aao.org/eye-health/diseases/low-vision-aids
  33. https://www.med.unc.edu/microscopy/wp-content/uploads/sites/742/2018/06/lm-ch-7-lenses.pdf
  34. http://www.phys.ufl.edu/courses/phy2054/fall08/GM_notes/chapter25_v1.pdf
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