# Steps for correcting chromatic aberrations in lens design

In many applications of lens design, avoiding chromatic aberration is a key performance requirement, in the video on post below we review some of the causes of chromatic aberration and methods to correct it.

•All optical media has dispersion, meaning the index of refraction depends on wavelength of spectrum

•The plots are not linear – they are often steeper in blue region and smoother in the red

•Approximation formulas for dispersion of glass include: Schott, Conrady, Herzberger, and Sellmeier.

A single optical element has chromatic difference in focus position for different wavelengths

Focusing error is Δs^′=-〖f^′〗_d/V_d

Where V_d=(n_d-1)/(n_F-n_C ) is Abbe number

n_d - index of refraction of the d-line of spectrum

n_F - index of refraction of the F-line of spectrum

n_C - index of refraction of the C-line of spectrum

Lenses with bigger Abbe number have smaller achromatic focus error

Chromatic difference depends on optical power and Abbe number only not on radii or shape of lens

To correct Chromatic aberrations, glasses with different V are needed. Usually it is “Crown glass” with higher V and “Flint glass” with lower V

To build achromatic doublet where 2 wavelength focus coincide, the next condition should be satisfied

Δs^′=-〖f^′〗_1/V_1 -〖f^′〗_2/V_2 =0

〖f′〗_1⁄〖f′〗_2 =-V_1⁄V_2

Therefore:

- Since V is positive for all optical glasses, then lenses have to be positive and negative.

- If doublet needs to have positive, then optical power of positive element has to be bigger than optical power of whole doublet

- To decrease optical power of positive element, Abbe number should be as big as possible

- Achromatic doublet can eliminate difference of couple of selected wavelengths but there is focus difference for other wavelengths. This difference names “secondary spectrum”

- The more difference in Abbe number of crown glass and flint glass the more secondary spectrum

Secondary spectrum is

〖∆S′〗_(F-d)=f′(P_1-P_2)/(V_1-V_2 )

Where P=(n_F-n_d)/(n_F-n_C )

Apochromatic correction means that the lens provides correction of chromatic aberration for 3 selected wavelengths for elimination of secondary spectrum.

Usually 3 lenses with 2-3 different glasses is used to achieve this performance.

The condition is P_1=P_2

For satisfying this condition unusual glasses with particular relations of V and P are required. Those are glasses with special dispersion behavior.

There are Crown glasses with long dispersion in blue region and Flint glass with short dispersion in blue region.

Schott in Germany invented such glasses.

They are named “Kurtz Flint Sonder” - KzFS type and “Lang Schwer Kron” – LgSK type glasses

The aim should be as small as possible “Tertiary spectrum”.

Super Apochromatic correction means that the lens provides correction of chromatic aberration for 4 selected wavelengths.

Usually 3-4 lens elements with 3 different glasses is used to achieve this performance.

Besides of Long Crown glass and Short Flint glass additional glass is used to get better result of design, for example Dense Flint – SF type

Chromatic difference of spherical aberration or spherochromatism can appear in achromatic corrected lens with large aperture.

As for glass achromatic doublet with 2 wavelength focus shift correction these conditions should be satisfied

Δs^′=-〖f^′〗_1d/V_1d -〖f^′〗_2d/V_2d =0

〖f′〗_1⁄〖f′〗_2 =-V_1⁄V_2

Abbe number for diffraction element is described as

V_d=λ_d/(λ_f-λ_c )=-3.452

Thus:

1) Diffractive optical element has essentially bigger dispersion than optical glass.

2) because Abbe number of diffractive element is negative then the Hybrid achromatic lens with positive optical power should be composed by positive refractive element and positive diffraction one. Therefore “crown element” of Hybrid achromatic doublet will have smaller optical power than whole achromatic lens.

Because essential dispersion of diffractive element “secondary spectrum” is essentially bigger than in full refractive achromatic doublet. Also secondary spectrum of Hybrid achromatic doublet is negative

Usually spherical aberration is corrected for one of wavelength of spectral waveband for example for green in visual band.

But some residual spherical aberration can be seen for other wavelengths inside of the working band despite of color correction.

Spherochromatism should be corrected additionally using additional parameters of lenses.Also there can be residual color difference for other aberrations: coma, field curvature and distortion.

7-element design with 4 types of glass. Wide band 5 color superachromat with correction of spherochromatizm

4 glass design was used for achieving 5 color correction superachromat in 0.4 to 0.9 microns waveband.

Spherochromatism is corrected using additional parameters provided by 7 elements design

Lateral color is chromatic aberration of principal rays.

Lateral color is difference of intersection of principal rays for blue and red light.

Lateral color if not corrected appears in field of view. If can be seen in simple eyepiece when look through the FOV from center to edge of field

## Summary

It is not possible to reach ideal color correction because of very different dispersion behavior of real existing glasses. Only correction for several wavelength of given wavelength region is possible.

The task is difficult for system with high aperture because correction of spherochromatizm going to be more difficult problem.

Lateral color is more difficult to eliminate and required additional parameters of optical system for correction.

But residul spectrum can be decreased below of diffraction limit which are result of diffraction in given optical system.