# Chromatic aberration

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In optics, chromatic aberration is caused by a lens having a different refractive index for different wavelengths of light (the dispersion of the lens).

Longitudinal and lateral chromatic aberration of a lens is seen as "fringes" of color around the image, because each color in the optical spectrum cannot be focused at a single common point on the optical axis.

Since the focal length f of a lens is dependent on the refractive index n, different wavelengths of light will be focused on different positions. Chromatic aberration can be both longitudinal, in that different wavelengths are focused at a different distance from the lens; and transverse or lateral, in that different wavelengths are focused at different positions in the focal plane (because the magnification of the lens also varies with wavelength).

## Minimizing chromatic aberrationEdit

There exists a point called the circle of least confusion, where chromatic aberration can be minimized. It can be further minimized by using an achromatic lens or achromat, in which materials with differing dispersion are assembled together to form a compound lens. The most common type is an achromatic doublet, with elements made of crown and flint glass. This reduces the amount of chromatic aberration over a certain range of wavelengths, though it does not produce perfect correction. By combining more than two lenses of different composition, the degree of correction can be further increased, as seen in an apochromatic lens or apochromat.

Many types of glass have been developed to reduce chromatic aberration, most notably, glasses containing fluorite. These hybridized glasses have a very low level of optical dispersion; only two compiled lenses made of these substances can yield a high level of correction.

The use of achromats was an important step in the development of the optical microscope.

An alternative to achromatic doublets is the use of diffractive optical elements. Diffractive optical elements have complementary dispersion characteristics to that of optical glasses and plastics. In the visible part of the spectrum, diffractives have an Abbe number of -3.5. Diffractive optical elements can be fabricated using diamond turning techniques.

### Mathematics of chromatic aberration minimizationEdit

For a doublet consisting of two thin lenses in contact, the Abbe number of the lens materials is used to calculate the correct focal length of the lenses to ensure correction of chromatic aberration. If the focal lengths of the two lenses for light at the yellow Fraunhofer D-line (589.2 nm) are f1 and f2, then best correction occurs for the condition:

$f_1 \cdot V_1 + f_2 \cdot V_2 = 0$

where V1 and V2 are the Abbe numbers of the materials of the first and second lenses, respectively. Since Abbe numbers are positive, one of the focal lengths must be negative, i.e. a diverging lens, for the condition to be met.

The overall focal length of the doublet f is given by the standard formula for thin lenses in contact:

$\frac{1}{f} = \frac{1}{f_1} + \frac{1}{f_2}$

and the above condition ensures this will be the focal length of the doublet for light at the blue and red Fraunhofer F and C lines (486.1 nm and 656.3 nm respectively). The focal length for light at other visible wavelengths will be similar but not exactly equal to this.

## Image processing to reduce chromatic aberrationEdit

Post-processing to remove chromatic aberration usually involves scaling the fringed colour channel, or subtracting some of a scaled version of the fringed channel.

## Black and white photographyEdit

Chromatic aberration also affects black and white photography. Although there are no colours in the photograph, chromatic aberration will blur the image. It can be reduced by using a narrow band colour filter, or by converting a single colour channel to black and white. This will however require longer exposure.