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Lenses

Lenses are used to "bend" light (i.e. to refract light). The result after the light passes through the lens is typically a magnified or minified image of the object illuminated.

Lenses are traditionally shaped as sections of spheres, such that the radius of curvature of the sphere of which they are comprised can describe the lens' curvature. The diagram of an eye and lens to the right demonstrates this concept.

Lenses show up in nature in the eyes of animals. They are used by humans to observe very tiny (microscope) and very far away (telescope) objects. They are also used in cameras, projectors, glasses, contact lenses, CD players, and other optical devices.

eye lens

Lens types

Below are two demonstrations of lenses: biconcave and biconvex. Several simple combinations are common (see figure on the right):

  • Converging: Biconvex, planoconvex, and positive meniscus lenses focus light.
  • Diverging: Biconcave, planoconcave, and negative meniscus lenses diverge light.

Meniscus lenses are the type used in eyeglass and contact lenses. Positive meniscus lenses are thicker at the center than the edges. The radius of curvature of the front surface is smaller than that of the back surface. It corrects hyperopia (farsightedness). Negative meniscus lenses are thinner at the center and wider at the edges. The radius of curvature of the front surface is larger than that of the back surface. It corrects myopia (nearsightedness).

lens types

Biconvex lens

1.8



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Biconcave lens

1.8



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Lensmaker's Equation

Equation to calculate the focal length of a lens in air:

$$P = \frac{1}{f} = (n-1)\left[\frac{1}{R_1}-\frac{1}{R_2}+\frac{d (n-1)}{n R_1 R_2}\right]$$:

(Focal length is measured from the center of the lens):

The power of the lens \(P\) (in diopters for f in meters) is equal to the inverse of the focal length, \(f\).:

The power is determined by the index of refraction, \(n\), of the lens material, the radius of curvature of the front surface, \(R_1\), the radius of curvature of the back surface, \(R_2\), and the thickness of the lens, \(d\), as measured along the axis (center).:

If the lens is thin enough, \(d \ll R_1 R_2\), then the "thin lens approximation" can be used::

$$P = \frac{1}{f} \approx (n-1)\left[\frac{1}{R_1}-\frac{1}{R_2}\right]$$:

For a collection of \(n\) thin lenses on the same axis, the focal length is just \(\frac{1}{f} \approx \frac{1}{f_1} + \frac{1}{f_2} + ... + \frac{1}{f_n}\):

Use the form below to calculate parameters in the lensmaker's equation based off the other parameters. Clicking on "calc" next to a field will calculate the value for that field given the values that have been put into the other fields.:

Diagram does not work with both \(R_1\) and \(R_2\) negative
Calculator works with any combination.


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? f is the focal length of the lens and the inverse of the power. If positive, the lens is convergent, and if negative, the lens is divergent.

? R1 is the radius of curvature of the front surface. By convention, if positive, the surface is convex, if negative it is concave.

link curvatures unlinked
? R2 is the radius of curvature of the back surface. By convention, if positive, the surface is concave, if negative it is convex.

? n is index of refraction. A vacuum has n=1. Typical lenses have values from 1.4 to 1.8

? d is the thickness of the lens at the axis (center).


Image properties

The focal length, \(f\) of a lens can described using the distance of the object to the lens, \(S_1\) and the distance of the lens to the image, \(S_2\):

$$ \frac{1}{f} = \frac{1}{S_1} + \frac{1}{S_2} $$

An object must be placed at \(S_1 \gt f\) for a sharp projected image. If the object is placed inside the focal length, a "virtual image" appears in front of the lens, and nothing is projected behind it. It is this virtual image that makes magnifying glasses useful.

The magnification of a lens can be described using the same parameters:

$$M = -\frac{S_2}{S_1} = \frac{f}{f-S_1} $$

If the absolute value of M is greater than 1, the image is magnified. If M is negative, the image is inverted.

Use the calculator below to find combinations of focus, object location, image location, and magnification.






Click and drag the arrow tip of the object (dark orange) to see the effect of object placement on the image. Click and drag either of the white focus points to see the effect of changing the focal length of the lens.


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If the image click-and-drag isn't working, try refreshing the page.

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