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Refractive index, complex
Complex refractive index of a medium (relative to vacuum) is defined as follows:
|m = m' - m" i||(1)|
where m' is the real part of the refractive index, m" is the imaginary part of the index and i = √(-1).
Complex refractive index of a medium depends primarily on the wavelength of light λ (usually given in vacuum), but also on the thermodynamic state (as defined by the temperature and pressure) of the medium, as well as on the concentration of solutes in that medium (such as the salts in seawater, see Refractive index of seawater, real part, formulas).
Refractive index, complex, and dielectric function
Complex refractive index, m = m' - m" i [where i = √(-1)] and the dielectric function, ε = ε' - ε" i of a medium are related as follows (for example, Fox M 2010, p. 7):
|ε = m 2||(1)|
|ε' = m' 2 - m" 2||(2)|
|ε" = 2 m' m"||(3)|
and (for example, Rakić A et al 1998)
|m' = 2-1/2 [ ( ε' 2 + ε" 2 )1/2 + ε' ]1/2||(4)|
|m" = 2-1/2 [ ( ε' 2 + ε" 2 )1/2 - ε' ]1/2||(5)|
The dielectric function is sometimes referred to as the dielectric constant. However, it is constant only at a specific frequency of the electromagnetic wave and a specific thermodynamic state (as defined by the temperature and pressure) of the material.
Refractive index, complex, imaginary part
The imaginary part of the complex refractive index is sometimes referred to as the extinction coefficient (for example, in a Wikipedia article on the refractive index) or as the absorption index. The extinction coefficient referred to in this context is not the same as the extinction coefficient related to the attenuation of light by a medium. Hence, the use of the term "extinction coefficient" in reference to the imaginary part of the refractive index may be confusing.
Refractive index, complex, imaginary part, and absorption of light
The imaginary part, m", of the complex refractive index, m, is related to the absorption of light by the medium. Indeed, the electric field, E, of a plane monochromatic electromagnetic wave propagating in the medium depends on distance, z, in the medium as follows:
|E||= E0 exp(-imkz)|
|= E0 exp(-im' kz) exp(-m"kz)||(1)|
where E0 is the electric field at z = 0, i = √(-1), k = 2π /λ is the wave number in vacuum (λ is the wavelength of light in vacuum).
The first exponential term in the second line of this equation describes a non-decaying oscillation of the wave amplitude in space. The second exponential term describes a decay of that oscillation with distance, z, in the medium, an effect of absorption of light by that medium.
The power of the eletromagnetic wave is proportional to the time average of the product E E*, where E* is the complex conjugate of E. Hence, the power of the wave decays in a medium with a non-zero imaginary part of the refractive index as exp(-2m"kz). On the other hand, the Lambert law of attenuation of light by a medium expresses this decay as exp(-az), where a is the absorption coefficient of the medium. By comparing the exponents, we obtain the following relationship between the imaginary part of the refractive index, m", and the absorption coefficient, a:
|m"(λ) = a(λ) λ / (4π)||(2)|
If the value of the imaginary part, m", of the refractive index in Eq. 1 is negative (in the present notation; see also Refractive index, complex, notations), then the amplitude of an electromagnetic wave increases as it propagates through the medium. Such a medium provides optical power gain.
Refractive index, complex, notations
There is a variety of notations for the complex refractive index, for example: m = n - ki, n = nr - nii, where i = √(-1).
Bohren CF and Huffman 1983 point out that the sign convention for the imaginary part of the refractive index is related to a choice of the sign of the exponent in the time-dependent part, exp(iωt), of the expression for an amplitude of a monochromatic electromagnetic wave. With exp(-iωt), one would have m = m' + m" i.
Refractive index, complex, real part
The real part, m' of the complex refractive index of a medium is a ratio of the velocity of light in the medium to that of a reference material. If that material is not named it is typically vacuum. In that case, m' may be referred to as the "absolute refractive index". In the optics of small particles, the refractive index of the particle material is generally taken to be relative to that of the medium surrounding the particle.
Refractive index, complex, real part, and molar refractivity
The real part of the complex refractive index of a material is related to the molecular refractivity, Am, of that material as follows (for example, Born M and Wolf 1999, p. 93):
|[ ( n' 2 - 1 ) / ( n' 2 + 2 ) ] ( W / ρ ) = Am||(1)|
where W is the molecular weight [g], ρ [g / cm3] is the density, and Am is defined as follows:
|Am = [( 4π ) / 3 ] Nm α||(2)|
where Nm ≅ 6.02 × 1023 is the Avogadro number (i.e. the number of molecules in a mole of the material) and α [cm3] is the mean plarizability of the molecules. To a "good approximation" (as stated by Born M and Wolf 1999, p. 94), the molar refractivity of a medium being a mixture to two media with molar refractivities Am1 and Am2, can be expressed as follows:
|Am = ( N1 Am1 + N2 Am2 ) / ( N1 + N2 )||(3)|
where N1 and N2 are the numbers of molecules of medium 1 and 2, respectively, per unit volume.
Eq. 1 is referred to as the Lorentz-Lorenz formula. Its derivation assumes that a molecule of the material reacts to the local electric field at the molecule's position. This local field is a sum of the electric field of the electromagnetic wave and the field created by all other molecules of the material assuming the uniform polarization of the material by the wave.
|[ ( ε' - 1 ) / ( ε' + 2 ) ] [ 3 / ( 4 π N ) ] = α||(4)|
where N [cm3] is the number of molecules per unit volume of the material.
Refractive index, meanings
The term "refractive index" is frequently used as short version of the terms "complex refractive index" and "real part of the complex refractive index".
Refractive index, of
carbon, see Refractive index of carbon
gold, see Refractive index of gold (includes references on the refractive index of other metals)
seawater, see Refractive index of seawater
silica: Kitamura R et al 2007 (review of data and formulas)
water, see Refractive index of water
Jonasz M. 2006. Refractive index (www.mjcopticaltech.com/Publications/RefInd.php).
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