This thread describes various "dispersion equations," which are used to calculate the refractive index of a lens material at different wavelengths. The refractive index in the blue end of the spectrum is actually higher than the refractive index in the red end. This leads to an optical effect known as chromatic dispersion, which results in the breaking up of white light into its component colors. Chromatic dispersion is responsible for lateral and axial chromatic aberration in spectacle lenses.
The Abbe value or constringence is a measure of chromatic dispersion in lens materials. The Abbe value (v) is calculated by comparing the mean refractive index of the lens across the visible spectrum to the difference in refractive indices between the blue and red ends of the spectrum:
where nd refers to the refractive index of the lens material for the helium d line at 587.56 nm, a wavelength produced by exciting helium gas, nF refers to the hydrogen F line at 486.13 nm, and nC refers to the hydrogen C line at 656.27 nm.
In some countries, the mercury e line at 546.07 nm is used as the reference wavelength for determining the mean refractive index, instead of the helium d line. (This is also why some manufacturers refer to certain high-index lens materials as "1.67" while others use "1.66.") Because of the effects of chromatic dispersion, the actual power of a lens or prism will depend upon the chosen reference wavelength. This concept is also important when measuring lenses using a device calibrated for either wavelength.
It is possible to estimate the refractive index of a lens material at a given wavelength, if you know the refractive index of the material at other wavelengths, using various dispersion equations. One common dispersion equation is Cauchy's equation, which has the form:
Of course, these dispersion equations require you to know the refractive index of the lens material at multiple wavelengths in order to determine the coefficients of the series (a, b, c...). Unfortunately, we are typically given only the Abbe value and the helium d refractive index at 587.56 nm.
A few months ago, I derived a "general" form of Cauchy's equation for any lens material, based on only the Abbe value (v) and helium d index (n), using the first two terms of the equation:
where the Greek lambda symbol represents the wavelength of interest. Only the first two terms from Cauchy's series can be used, since any additional terms would require more simultaneous equations than we could solve using only the Abbe value and refractive index.
So, using this equation, we can "estimate" that the refractive index of the blue hydrogen line (at 486.13 nm) in polycarbonate is:
While the refractive index of the red hydrogen line (at 656.27 nm) is:
Similarly, we can estimate that the mean refractive index of the lens material for surfacing or power calculations using the mercury e line (at 546.07 nm), instead of the standard helium d line used in the US, with this equation:
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