Difference between revisions of Diopters
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This is the resulting equation at the beginning of the article. It also explains why the focal power is increased for objects at closer distances: mainstream optometry calls this the "add" for presbyopia, although they typically use the minimum amount required for you to see at 40 cm with full distance correction using accommodation. For example, if you choose 80 cm as the working distance for your [[differentials]] (resulting in a +1.25 dpt "add"), and your blur horizon is 50 cm (resulting in -2 dpt), the formula is | This is the resulting equation at the beginning of the article. It also explains why the focal power is increased for objects at closer distances: mainstream optometry calls this the "add" for [[presbyopia]], although they typically use the minimum amount required for you to see at 40 cm with full distance correction using accommodation. For example, if you choose 80 cm as the working distance for your [[differentials]] (resulting in a +1.25 dpt "add"), and your blur horizon is 50 cm (resulting in -2 dpt), the formula is | ||
<math>\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{80\ cm}+\frac{1}{-50\ cm}=1.25\ dpt + \left(-2\ dpt\right) = -0.75\ dpt</math> | <math>\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{80\ cm}+\frac{1}{-50\ cm}=1.25\ dpt + \left(-2\ dpt\right) = -0.75\ dpt</math> | ||
=== Cylinder === | === Cylinder === | ||
A cylindrical lens of power | A cylindrical lens of focal power P<sub>cyl</sub> has power P at angle θ from its axis: | ||
<math> | <math>P = P_{cyl} (\sin\theta)^2</math> | ||
==== Axis ==== | |||
Axis is usually in degrees modulo 180. It is popular for 0 to be written as 180 in some areas. | |||
==== Transposition ==== | ==== Transposition ==== | ||
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<math> | <math>P = P_{cyl} \left( 1 - (\cos\theta)^2 \right) = P_{cyl} + \left(-P_{cyl}\right)(\sin{\left(\theta - 90^{\circ}\right)})^2</math> | ||
We can see that by adding the cylindrical power to the spherical power and inverting the cylinder and adding 90 degrees (same as subtracting 90 degrees, since the axis is in modulo 180 degrees) to the axis, we get an equivalent combination. | We can see that by adding the cylindrical power to the spherical power and inverting the cylinder and adding 90 degrees (same as subtracting 90 degrees, since the axis is in modulo 180 degrees) to the axis, we get an equivalent combination. | ||
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By calculating the average value over all angles using an integral, the result<ref>it is only necessary to integrate one or two periods: https://www.wolframalpha.com/input/?i=average+of+%28sin+x%29%5E2+from+0+to+2+pi</ref> is | By calculating the average value over all angles using an integral, the result<ref>it is only necessary to integrate one or two periods: https://www.wolframalpha.com/input/?i=average+of+%28sin+x%29%5E2+from+0+to+2+pi</ref> is | ||
<math> | <math>P_{avg} = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} P_{cyl} (\sin{t})^2 \,dt = \frac{1}{2} P_{cyl}</math> | ||
This is why the spherical equivalent has power equal to half of the cylinder's power. | This is why the spherical equivalent has power equal to half of the cylinder's power. | ||
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<math> | <math>P = P_{cyl} (\sin{\left(\theta + \phi\right)})^2 = P_{cyl} \left( \frac{1-\cos{\left(2\theta + 2\phi\right)}}{2} \right) = \frac{1}{2} P_{cyl} + \frac{-P_{cyl}}{2} \cos{\left(2\theta + 2\phi\right)}</math> | ||
The constant parts are added with the spherical components. The cosines can be added by converting them to [https://en.wikipedia.org/wiki/Phasor phasors] and adding the phasors together. The resulting phasor corresponds to one of two cylindrical lenses (see the section on Transposition), and its corresponding spherical component must be subtracted from the total spherical component. | The constant parts are added with the spherical components. The cosines can be added by converting them to [https://en.wikipedia.org/wiki/Phasor phasors] and adding the phasors together. The resulting phasor corresponds to one of two cylindrical lenses (see the section on Transposition), and its corresponding spherical component must be subtracted from the total spherical component. | ||
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Induced prism can be calculated using Prentice's rule. Similar to Vertex Distance, decentration is less of an issue for smaller power lenses. | Induced prism can be calculated using Prentice's rule. Similar to Vertex Distance, decentration is less of an issue for smaller power lenses. | ||
The amount of prism P induced by decentration c of a lens of power f is | The amount of prism power P induced by decentration c of a lens of power f is | ||
<math>P=cf</math> | <math>P=cf</math> |
Revision as of 16:18, 28 March 2022
Diopter is a measure of the optical power P of a lens (or mirror) and is equal to the reciprocal of focal length in meters. The most common unit symbol for diopters is dpt, D, or m-1.
- In EM, we use the cm measurement to calculate diopters needed to correct refraction of the eye. If you can see 50 cm clearly, your diopters will be .
- Lenses in series add their powers: if you're wearing -2 diopter contacts (adjusted for glasses strength) and put +1 diopter reading glasses over the contacts you're in effect wearing -1 diopters.
- There are some caveats such as vertex distance, since moving the lens away effectively gives you a weaker negative lens or stronger positive lens. There is also decentration, which induces prism when the lens is moved to the side. These effects become negligible for weaker lenses.
- According to the thin lens sign convention, negative focal power is diverging, and positive focal power is converging.
- A lens with a negative diopter sign compensates for nearsightedness while a lens with a positive diopter sign compensates for farsightedness.
0.00 to -0.50 dpt | Not really considered myopic, probably don't need glasses |
-0.50 to -1.00 dpt | Mild myopia, normalized sometimes not needed |
-1.00 to -2.50 dpt | Mild myopia, differentials possibly not needed |
-2.50 to -3.00 dpt | Mild myopia, differentials probably needed |
-3.00 to -6.00 dpt | Moderate myopia, glasses always needed |
-6.00 to -10.00 dpt | High myopia |
-10.00+ dpt | Very high myopia. Field of view significantly reduced. |
Gap and ratio
Comparisons between two diopters is typically expressed using one of these terms:
- diopter gap (or diopter difference): absolute difference in diopters between two values
- diopter ratio: ratio of one diopter value to another (such as right eye / left eye)
For example, consider the following correction:
OD: -1.5 SPH / -1.5 CYL OS: -1.0 SPH / -2.0 CYL
It can be expressed as a 0.5 dpt gap in both SPH and CYL, a 1.5 ratio in SPH and a 0.75 ratio in CYL:
|(-1.5 dpt) - (-1.0 dpt)| = 0.5 dpt |(-1.5 dpt) - (-2.0 dpt)| = 0.5 dpt (-1.5 dpt) / (-1.0 dpt) = 1.5 (-1.5 dpt) / (-2.0 dpt) = 0.75
In EM articles, the term diopter ratio is often used interchangeably for left-right diopter gap[1], for example when talking about reducing a correction while keeping the gap the same. This can also be expressed as a percentage difference between the two diopter values[2] (e.g. the 0.5 dpt difference between the right and left eyes here is equivalent to 0.5 dpt / |-1.5 dpt| = 0.33 or 33%).
The general recommendation is that the left-right diopter gap is constant across all lenses being used. However, some old EM articles show successful cases where differentials are equalized but normalized have a 0.25 D gap.[3]
Confusingly, diopter gap is also sometimes used to refer to the diff-norm gap, the difference between differentials and normalized or the spherical equivalent of that difference.[4]
It is often useful to disambiguate what is being compared:
- left-right gap: left eye diopters minus right eye diopters, ignoring axis
- In the example above, the left-right gap is +0.5 SPH -0.5 CYL.
- The axis is ignored, and the cylinder powers are subtracted without using lens combination calculations.
- diff-norm gap: differentials diopter minus normalized diopters, ignoring axis
- For example, if normalized is -2 SPH -0.5 CYL, and differentials are -0.75 SPH, the diff-norm gap is 1.25 SPH 0.5 CYL or 1.5 SPH equivalent.
- The axis is ignored.
- This quantity is generally positive, as more plus sphere is needed for close-up than distance vision.
Technical Details
This section is for the math-savvy people. It explains concepts in more detail, but knowledge of it is not strictly necessary to use the EM method.
Thin Lens Equation
The focal length of a lens is given by the lensmaker's equation. By assuming the lens is much thinner than the radius of curvature, therefore assuming lens thickness is zero, we get a simplified version of the lensmaker's equation. We can do some further derivation, we arrive at the thin-lens equation:[5]
According to the thin lens sign convention,
- di is positive if it is a real image on the opposite side of the lens as the object, and it is negative if it is a virtual image on the same side of the lens as the object.
- f is positive for converging lens and negative for diverging.
This is also sometimes presented in the Newtonian form:
Examples
"Full correction" takes an object at infinity and produces a virtual image at your far point distance d:
This is the resulting equation at the beginning of the article. It also explains why the focal power is increased for objects at closer distances: mainstream optometry calls this the "add" for presbyopia, although they typically use the minimum amount required for you to see at 40 cm with full distance correction using accommodation. For example, if you choose 80 cm as the working distance for your differentials (resulting in a +1.25 dpt "add"), and your blur horizon is 50 cm (resulting in -2 dpt), the formula is
Cylinder
A cylindrical lens of focal power Pcyl has power P at angle θ from its axis:
Axis
Axis is usually in degrees modulo 180. It is popular for 0 to be written as 180 in some areas.
Transposition
We can understand why there are two different ways to write a spherical and cylindrical lens combination, using the Pythagorean trigonometric identity and the complementary angle identity:
We can see that by adding the cylindrical power to the spherical power and inverting the cylinder and adding 90 degrees (same as subtracting 90 degrees, since the axis is in modulo 180 degrees) to the axis, we get an equivalent combination.
For example, -1 sph -1 cyl 1 axis is the same as -2 sph +1 cyl 91 axis.
In general, optometrists prefer to use a negative cylinder and ophthalmologists prefer to use a positive cylinder, but the two forms are equivalent.
Spherical Equivalent
By calculating the average value over all angles using an integral, the result[6] is
This is why the spherical equivalent has power equal to half of the cylinder's power.
Adding/Combining Lenses
Multiple lens, each with spherical and cylindrical components (not necessarily at the same axis) can be added to form one lens with a spherical and cylindrical component.
We can use the double angle formula to convert each cylindrical lens into a constant plus a cosine:
The constant parts are added with the spherical components. The cosines can be added by converting them to phasors and adding the phasors together. The resulting phasor corresponds to one of two cylindrical lenses (see the section on Transposition), and its corresponding spherical component must be subtracted from the total spherical component.
There are implementations of this at
Decentration
Induced prism can be calculated using Prentice's rule. Similar to Vertex Distance, decentration is less of an issue for smaller power lenses.
The amount of prism power P induced by decentration c of a lens of power f is
1 prism diopter displaces 1 cm for an object 1 m away. If c is in cm, and f is in diopters, then P is in prism diopters.
A prism with apex angle a and refractive index n results in angle of deviation of the light d, which is equal to P prism diopters:
Vertex Distance
See Vertex distance#Calculation
References
- ↑ The EndMyopia Blog, https://endmyopia.org/the-diopter-ratio-trap-dont-favor-one-eye/
- ↑ The EndMyopia Blog, https://endmyopia.org/reducing-diopter-ratio-diy-patching-solution-pro-topic/
- ↑ https://endmyopia.org/progress-improving-centimeter-62-90/ and https://endmyopia.org/saras-journey-truth-long-term-vision-improvement-potential/
- ↑ https://endmyopia.org/pro-topic-managing-your-maximum-diopter-gap/
- ↑ see derivations at https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/02%3A_Geometric_Optics_and_Image_Formation/2.05%3A_Thin_Lenses
- ↑ it is only necessary to integrate one or two periods: https://www.wolframalpha.com/input/?i=average+of+%28sin+x%29%5E2+from+0+to+2+pi