# Dioptric Power

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: the inverse of the focal length

Alternative terms: power, focusing power

Units: m^{−1}

Formula symbol: <$\varphi$>

The dioptric power of a focusing or defocusing optical element is defined as the inverse of its effective focal length. Sometimes it is just called *power*, but this is ambiguous because *power* occurs with various different meanings, e.g. magnification or optical power.

The dioptric power is measured in units of m^{−1}, also called *diopters* (dpt). It is common to specify the dioptric power of prescription glasses, for example, whereas the focal length is usually specified for standard lenses, microscope objectives and photographic objectives.

In many cases, the dioptric power is a more natural quantity than the focal length because stronger focusing action implies a higher dioptric power but a shorter (smaller) focal length. For example, the dioptric power of the thermal lens in a laser crystal is proportional to the dissipated power. The width of the stability zones of a laser resonator with respect to dioptric power of the thermal lens depends only on the minimum mode radius in the laser crystal and on the optical wavelength, whereas the stability range in terms of focal length has a more complicated dependence.

Some additional aspects come into play when the medium before and after the optical system is not the same. For example, we may consider an underwater camera, where the object side is filled with water, while the interior of the camera is filled with air. In that situation, the front focal length of the camera objective is larger than the back focal length. The dioptric power must be calculated from the effective focal lengths, which is related to the case <$n = 1$>. One thus takes the inverse of the back focal length, which is the same as the refractive index of water divided by the front focal length.

The dioptric power of a single interface between two media is the difference of refractive index divided by the radius of curvature of the surface.

The combination of two thin lenses at a close distance has a dioptric power which is simply the sum of the powers of the two lenses. If the distance <$d$> is not negligible, the total dioptric power is:

$$\phi = {\phi _1} + {\phi _2} - {\phi _1}\;{\phi _2}\;d$$## Example: Dioptric Power of Prescription Glasses

Simple calculations can be done concerning the required dioptric power of prescription glasses under different circumstances.

A well functioning human eye can view distant objects with the relaxed eye's lens, but can also accommodate to shorter viewing distances by increasing the dioptric power of the eye. For a minimum viewing distance of 20 cm, for example, the dioptric power needs to be increased by 1 / 20 cm = 5 dpt. Considered in the context of geometrical optics, this is because 5 dpt are required for transforming rays which originate from an object in a distance of 20 cm into parallel rays (as would be obtained for a very distant object).

With increasing age, the eye more and more loses its capability to accommodate to different distances, i.e., it can vary its dioptric power only in a rather limited range of e.g. 1 dpt width. If the relaxed eye can comfortably see distant objects, for reading text in the distance of only 50 cm, for example, one will require prescription glasses with 1 / 50 cm = 2 dpt, well beyond the remaining accommodation power of the old eye. For reading text on a computer monitor in a more convenient distance of 80 cm, one requires only 1 / 80 cm ≈ +1.2 dpt. However, it may be that the relaxed eye already requires some correction (some positive or negative dioptric power) to comfortably view distant objects, and that correction would also have to be added for reading at shorter distances.

In practice, the situation can become more complicated for various reasons:

- The eye may have some astigmatism, which implies that it requires different dioptric powers in different directions. Only for sufficiently weak astigmatism, an ordinary spherical lens will provide good enough vision.
- The two eyes may require different corrections.
- The distance between the lens centers for the two eyes should be appropriately adjusted because otherwise one obtains an additional change of viewing direction. That aspect becomes more critical for glasses with large dioptric power.

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2023-01-22

Is there a way to measure the dioptric power of a laser crystal inside the laser resonator?

The author's answer:

There are ways to do that, but it generally tricky to do. For example, one may send a test laser beam through the crystal – at a wavelength where the resonator mirrors are transparent –, and measure changes of beam radius at some later position. Or one finds the stability limit of the resonator when moving an end mirror; results can be compared with a resonator model.