Opacity (optics)

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Opacity is the measure of impenetrability to electromagnetic or other kinds of radiation, especially visible light. In radiative transfer, it describes the absorption and scattering of radiation in a medium, such as a plasma, dielectric, shielding material, glass, etc. An opaque object is neither transparent (allowing all light to pass through) nor translucent (allowing some light to pass through). When light strikes an interface between two substances, in general some may be reflected, some absorbed, some scattered, and the rest transmitted (also see refraction). An opaque substance transmits very little light, and therefore reflects, scatters, or absorbs most of it. Both mirrors and carbon black are opaque. Opacity depends on the frequency of the light being considered. For instance, some kinds of glass, while transparent in the visual range, are largely opaque to ultraviolet light. More extreme frequency-dependence is visible in the absorption lines of cold gases. In general, a material tends to emit light in the same proportions as it absorbs it.

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Definition

The opacity κν gives the rate of absorption (or extinction), which is the fraction of the intensity Iν, of the radiation that is absorbed or scattered per unit distance along a ray of propagation:

{\partial I_\nu\over\partial x}=-I_\nu\kappa_\nu

For a given medium it has a numerical value that may range between 0 and infinity. It is also called the absorption coefficient (see also extinction coefficient). In general κν depends on the frequency ν of the radiation, as well as the density, temperature, and composition of the medium. The mean free path is the distance a photon travels in the medium before absorption or scattering is defined as 1 / (κνρ), where ρ is the density of the material. The notation κλ is the opacity described as a function of wavelength λ. While many materials are very opaque (steel in visible light having near-infinite opacity), and others very transparent (air in visible light having near-zero opacity), so that opacity often seems to be a boolean property, many others (such as water) have intermediate opacity.

In astronomy and planetary imaging fields, tau, the optical depth, defines the opacity: zero indicates transparent and higher numbers indicate more and more opaque in an inverse exponential fashion, for example a tau of 1 indicates 36 percent of the light passes (e-1 = 0.36), and a Tau of 5 indicates less than 1 percent passes (e -5 = 0.0067).[citation needed]

\kappa_\nu^{-1}
\frac{1}{\kappa} = \frac{\int_0^{\infty} \kappa_{\nu}^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}
\kappa_{\rm ff}(\rho, T) = 0.64 \times 10^{23} (\rho[ {\rm g}~ {\rm cm}^{-3}])(T[{\rm K}])^{-7/2} {\rm cm}^2 {\rm g}^{-1}
\frac{1}{\kappa} = \frac{\int_0^{\infty} (\kappa_{\nu, {\rm es}} + \kappa_{\nu, {\rm ff}})^{-1} u(\nu, T) d\nu }{\int_0^{\infty} u(\nu,T) d\nu}

Applications

Rough plot of Earth's atmospheric opacity to various wavelengths of electromagnetic radiation. The human eye has evolved so as to be sensitive to a spectrum of low opacity (high transmittance), the "optical window". Rough plot of Earth's atmospheric opacity to various wavelengths of electromagnetic radiation. The human eye has evolved so as to be sensitive to a spectrum of low opacity (high transmittance), the "optical window".

In astrophysics, the variations in opacity within a star are important to the understanding of radiation transfer in stellar atmospheres and the spectra we observe.

In several types of chemical analysis, the concentration of a sample in a transparent medium (typically air or water) is determined via measuring its opacity or absorbance. In spectrophotometry the device identifies the sample's constituent substances from their absorbances.

Opacity is also used as a measurement of particulate emissions.

Extinction coefficient

The extinction coefficient for a particular substance is a measure of how well it scatters and absorbs electromagnetic radiation (EM waves). If the EM wave can pass through very easily, the material has a low extinction coefficient. Conversely, if the radiation hardly penetrates the material, but rather quickly becomes "extinct" within it, the extinction coefficient is high.

A material can behave differently for different wavelengths of electromagnetic radiation. Glass is transparent to visible light, but many types of glass are opaque to ultra-violet wavelengths. In general, the extinction coefficient for any material is a function of the incident wavelength. The extinction coefficient is used widely in ultraviolet-visible spectroscopy. See below for its exact definition.

Physical definitions

Main article: Mathematical descriptions of opacity

The parameter used to describe the interaction of electromagnetic radiation with matter is the complex index of refraction, ñ, which is a combination of a real part and an imaginary part:

\tilde{n}=n-ik.

Here, n is also called the index of refraction, which sometimes leads to confusion. k is the extinction coefficient, which represents the damping of an EM wave inside the material. Both depend on the wavelength.

An EM wave travels in the material with velocity v and angular frequency ω. The time-varying electric field of the wave is described by

\mathbf{E}(z,t) = \mathbf{E}_0 e^{i\omega(t - \frac{z}{v})},
\mathbf E

The index of refraction is defined to be the ratio of the speed of light in a vacuum to the speed of the EM wave in the medium:

\tilde{n} = \frac{c}{v}.
\tilde{n}
\frac{1}{v} = \frac{n}{c} - i\frac{k}{c}.

Substituting this in the expression for the EM wave's electric field gives

\mathbf E(z, t) = \mathbf E_0 e^{i\omega(t - z(\frac{n}{c}))} e^{-(\frac{k \omega}{c})z}.

This expression describes a propagating electromagnetic wave with an exponentially damped amplitude due to the k term. This term causes the EM wave to "die out" as it travels further into the material. The intensity of the wave, which corresponds to the energy it carries with it, is simply the square of the magnitude of the wave's electric field. The intensity of the wave is therefore

I(z) = I_0 e^{-\frac{2\omega k}{c}z}.

A law called the Beer-Lambert law states that in any medium that is absorbing light, the decrease in intensity I per unit length z is proportional to the instantaneous value of I. In mathematical form this is

\frac{dI\left( z \right)}{dz}={-\alpha I\left(z\right)},

where α is the absorption coefficient of the material for that wavelength of EM radiation. This equation has the solution

{I\left(z \right)}={I_0 e^{-\alpha z}}

where I0 is the intensity of the electromagnetic radiation at the surface of the absorbing medium. Comparing the two expressions for intensity obtained above gives

\alpha = \frac{2\omega k}{c}.

Since c here denotes the speed of the EM wave in vacuum,

c= \frac{\omega}{2\pi}\lambda

Substituting this in the expression above and rearranging shows that the extinction coefficient and the absorption coefficient are related by

k={\frac{\lambda}{4\pi}}\alpha

where λ is the vacuum wavelength (not the wavelength of the EM wave in the material).