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| {{Distinguish|Cauchy's functional equation}}
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| [[Image:Cauchy-equation-1.svg|thumb|Refractive index vs. wavelength for [[BK7 glass]]. Red crosses show measured values. Over the visible region (red shading), Cauchy's equation (blue line) agrees well with the measured refractive indices and the Sellmeier plot (green dashed line). It deviates in the ultraviolet and infrared regions.]]
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| '''Cauchy's equation''' is an [[empirical relationship]] between the [[refractive index]] and [[wavelength]] of light for a particular [[transparency (optics)|transparent]] material. It is named for the mathematician [[Augustin-Louis Cauchy]], who defined it in 1836.
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| ==The equation==
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| The most general form of Cauchy's equation is
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| :<math> n(\lambda) = B + \frac {C}{\lambda^2} + \frac{D}{\lambda^4} + \cdots,</math> | |
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| where ''n'' is the refractive index, λ is the wavelength, ''B'', ''C'', ''D'', etc., are [[coefficient]]s that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for λ as the [[vacuum wavelength]] in [[micrometre]]s.
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| Usually, it is sufficient to use a two-term form of the equation:
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| :<math> n(\lambda) = B + \frac{C}{\lambda^2},</math>
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| where the coefficients ''B'' and ''C'' are determined specifically for this form of the equation.
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| A table of coefficients for common optical materials is shown below:
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| {| border="1" cellspacing="0" cellpadding="4"
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| |'''Material''' || '''B''' || '''C''' (μm<sup>2</sup>)
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| |Fused silica || 1.4580 || 0.00354
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| |Borosilicate glass BK7 || 1.5046 || 0.00420
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| |Hard crown glass K5 || 1.5220 || 0.00459
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| |Barium crown glass BaK4 || 1.5690 || 0.00531
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| |Barium flint glass BaF10 || 1.6700 || 0.00743
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| |Dense flint glass SF10 || 1.7280 || 0.01342
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| |}
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| The theory of light-matter interaction on which Cauchy based this equation was later found to be incorrect. In particular, the equation is only valid for regions of normal [[dispersion (optics)|dispersion]] in the [[visible]] wavelength region. In the [[infrared]], the equation becomes inaccurate, and it cannot represent regions of anomalous dispersion. Despite this, its mathematical simplicity makes it useful in some applications.
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| The [[Sellmeier equation]] is a later development of Cauchy's work that handles anomalously dispersive regions, and more accurately models a material's refractive index across the [[ultraviolet]], visible, and infrared spectrum.
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| ==References==
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| *F.A. Jenkins and H.E. White, ''Fundamentals of Optics'', 4th ed., McGraw-Hill, Inc. (1981).
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| ==See also==
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| *[[Sellmeier equation]]
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| [[Category:Optics]]
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| [[Category:Equations]]
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