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In general, the refractive index of a certain medium tends to decrease with increasing wavelength.

But are there any materials in which the opposite phenomenon occurs? That is, in which the refractive index increases with increasing wavelength (which would lead to an "inverted" spectrum, if a light beam reached said material)?

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In general, the refractive index of light increases with the frequency of light (decreases with increasing wavelength, which is called "normal dispersion". There are, however, materials, where the refractive index decreases with the frequency of light ("abnormal or anomalous dispersion"). According to the Kramers-Kronig Relations, the frequency dependence of the refractive index is related to the frequency dependence of the light absorption in a material. Most optical media have the strongest absorption band in the ultra-violett range. This gives the normal increase of refractive index with frequency. When you measure the dispersion of the refractive index in a material above a strong absorption band, you can ( in a certain frequency range) get the "abnormal (anomalous) dispersion", i.e., a decrease of refractive index with increasing light frequency.

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