Reflectivity ρ is the ratio of the radiation flux Φ_{r} reflected by a sample surface to the incident radiation flux Φ_{i}:

Sometimes the term "reflectivity" is understood as the ratio of the mentioned fluxes when the sample reflects volumetrically including its interior if it is semitransparent to thermal radiation. In this case the reflection depends on sample thickness and instead of "reflectivity" we may recommend the use of the term "reflection coefficient". The methods for calculation and measurement of reflectivity are well developed.

For optically smooth sample surface (if its mean square microroughness is at least 100 times less than the wavelength of the radiation) the reflection is specular and the angle of reflection is equal to the angle of incidence. The value of reflectivity as a function of an incident angle may be calculated using Fresnel's formulas if refractive index n and absorption index χ of sample material are known.

In spectral range of transparency of two adjacent media (1 and 2), when the absorption indexes x_{1} and x_{2} are small in comparison with refractive indexes n_{1} and n_{2}, the following expressions are applied for the perpendicular () and parallel (||) polarized components of incident radiation:

where n_{21} = n_{2}/n_{h} and θ is an incident angle of radiation from first medium onto the second relative to the surface normal.

For incident natural unpolarized radiation the parallel and perpendicular components have an equal intensity and the mean arithmetic value for this radiation may be taken as reflectivity:

The angular dependence of reflectivity in this case has the form shown in Figure 1. Curve 1 is related to reflectance of natural nonpolarizated radiation, curve 2 is related to ρ_{} and curve 3 — to ρ_{||}. At n_{21} > 1 the perpendicular component is increased monotonically to unity with increasing angle θ, and the parallel component first decreases to zero and then increases to unity. The parallel component is equal to zero at the Bruster angle θ_{Br} (Figure 1a).

In the case of incidence of radiation from an optically more dense medium onto an interface with an optically less dense one (n_{21} < 1) the value of Bruster angle is lower, and starting from some value of so-called critical incident angle θ_{c} = sin^{–1} (n_{1}/n_{2}) at θ > θ_{c} the full internal reflection takes place, i.e., ρ_{} — ρ_{||} = 1 (Figure 1b).

In radiation transfer the case is often found when the medium 1 is nonabsorbing (χ_{1} = 0) and medium 2 is absorbing. In the more general case the medium 2 may have both a transparency region of the spectrum (χ_{2} = 0) and a nontransparency one (χ_{2} > 0). The expressions for the reflectivity components in spectral region of nontransparency of the second medium will have a form:

where

For reflection of medium 2 in vacuum, air or another small density gas it may be taken that refractive index n_{1} is equal to unity.

In Figure 2 the angular dependence of polarized components of reflectivity of the surface between medium 2 and medium 1 (with n_{1} = 1) is shown. The curves referred to the perpendicular polarized component of radiation are marked by one stroke, the curves marked with two strokes are referred to the parallel component.

The curves 1 describe the data for nonabsorbing medium with n_{2} = 1.5, χ_{2} = 0, the curves 2 — for n_{2} = 1.5, χ_{2} = 1.0, the curves 3 — for n_{2} = 11, χ_{2} = 6. It may be seen that the component ρ_{} has a minimum at large incident angles and this minimum is the deeper the greater the value of χ_{2}.

For radiation incident along the normal to the interface (θ = 0) the expression for reflectivity has a form:

However, for most important practical cases, the reflection surface (interface) is not optically smooth, the reflection is not specular and the reflected energy has a distribution of directions over a hemisphere whose base is the reflecting surface. Here other parameters describing the reflection must be introduced. The two-directional (bidirectional or directional-directional) reflectivity describes the ratio of the radiation intensity reflected in the direction determined by the unit vector to the radiation flux falling on unit area of a surface in an elemental solid angle dΩ from another direction determined by the unit vector , oriented under some angle to the external normal :

In this formula is the intensity of the incident radiation. The value of two directional reflectivity has the dimension of inverse solid angle (Sr^{–1}).

The directional-hemispherical reflectivity is the ratio of the hemispherically reflected (in all directions determined by the unit vector ) radiation flux to the radiation flux falling on the sample surface in an elemental solid angle dΩ from the direction determined by the unit vector oriented at some angle to the external normal :

The hemispherical-directional reflectivity is the ratio of the intensity of radiation reflected in some direction to the hemisphere-averaged radiation flux falling on unit area of surface from all hemispherical directions :

The two hemispherical (hemispherical-hemispherical) reflectivity ρ(h, h) is the ratio of the radiation flux reflected in the hemisphere in all directions to the radiation flux incident from all directions :

The directional reflection characteristics depend on the optical properties of the substance, temperature, wavelength, sizes, and the geometric shape of the surface microroughness.

The case is widely encountered where the microroughness sizes are comparible with the wavelength. In this case, theoretical calculations of directional reflection characteristics cannot be acheived and the characteristics have to be determined experimentally. At present the quantity of experimental data is insufficient. Basically there are the data on normal-hemispherical reflectivity at room temperature. The study of two directional reflectivity is very laborious. In this case many experimental points need to be measured. There are practically no such studies.

The two directional reflectivities are used in radiation transfer calculations as a rule only for specular reflection. If we are dealing with rough surfaces, taking into account the dependence of reflection (and emission) characteristics on direction is very difficult in most cases even for very simple shapes of radiating surfaces. Generally in this case the more simple model of diffuse (not dependent on angle) reflection and emission of the rough surface is used. In a case where the contribution of radiation directed at big angles to the surface normal is rather large it is possible to employ approximate methods of radiation transfer calculations. For example, it may be possible to use a model taking into account the directional emission by considering reflectivity as the sum of diffuse and specular components.

#### REFERENCES

Hsia, J. J., Richmond, J. C. (1976) Bidirectional reflectometry, 1. A high resolution laser bidirectional reflectometer with results of several optical coatings, *J. Res. Nat. Bur. Stand*. A., 80–2, 189–205, 1976.

Seigel, R., HoweJl, J. R. (1992) *Thermal Radiation Heat Transfer*, 3rd edn., Washington D. C, Hemisphere, 1992.

Toulukian, Y. S., De Witt, D. P. (1972) Thermophysical properties of matter, *Thermal Radiation Properties*, Vol. 7, *Metallic Elements and Alloys*, 1970, Vol. 8, Nonmetallic solids, 1972, New York—Washington, 1FI—Plenum.

#### References

- Hsia, J. J., Richmond, J. C. (1976) Bidirectional reflectometry, 1. A high resolution laser bidirectional reflectometer with results of several optical coatings,
*J. Res. Nat. Bur. Stand*. A., 80â€“2, 189â€“205, 1976. - Seigel, R., HoweJl, J. R. (1992)
*Thermal Radiation Heat Transfer*, 3rd edn., Washington D. C, Hemisphere, 1992. - Toulukian, Y. S., De Witt, D. P. (1972) Thermophysical properties of matter,
*Thermal Radiation Properties*, Vol. 7,*Metallic Elements and Alloys*, 1970, Vol. 8, Nonmetallic solids, 1972, New Yorkâ€”Washington, 1FIâ€”Plenum.