- Importance of effective emissivity calculation
- Reflection Models
- Cavity Shapes
- Viewing Conditions
- Temperatures
- Monte Carlo Ray Tracing
At the design stage, calculation is the only way to evaluate the quality of the blackbody, to determine its optimal parameters, and to choose suitable material. But if you bought a blackbody calibration source and intend to use the value of effective emissivity specified by the manufacturer, you should be doubly careful because:
1) Values of effective emissivity might be correct only for a part of operational spectral range.
2) None of blackbody manufacturers never purchased our blackbody emissivity modeling software. Other analogous software simply does not exist.
3) If effective emissivities were measured, you may not know which measured equipment was used and what is the precision of such measurements.
Measurements performed by independent researchers are not always confirm manufacturer's values. The picture at left shows how manufacturer's emissivity of a flat-plate blackbody agrees with that measured at NIST (adopted from: S. N. Mekhontsev, V. B. Khromchenko, L. M. Hanssen, "NIST Radiance Temperature and Infrared Spectral Radiance Scales at Near-Ambient Temperatures," Int. J. Thermophys. 29, 1026-1040 (2008) - Fig. 14. Effective spectral directional emissivity of a high-temperature flat-plate BB. Manufacturer specification is 0.95 (+0.00, -0.05)).
If such disagreement takes place for flat-plate blackbody (metallic plate coated with black paint) for which emissivity measurement is not too difficult, one can imagine the situation with the cavity radiators.
1) Values of effective emissivity might be correct only for a part of operational spectral range.
2) None of blackbody manufacturers never purchased our blackbody emissivity modeling software. Other analogous software simply does not exist.
3) If effective emissivities were measured, you may not know which measured equipment was used and what is the precision of such measurements.
Measurements performed by independent researchers are not always confirm manufacturer's values. The picture at left shows how manufacturer's emissivity of a flat-plate blackbody agrees with that measured at NIST (adopted from: S. N. Mekhontsev, V. B. Khromchenko, L. M. Hanssen, "NIST Radiance Temperature and Infrared Spectral Radiance Scales at Near-Ambient Temperatures," Int. J. Thermophys. 29, 1026-1040 (2008) - Fig. 14. Effective spectral directional emissivity of a high-temperature flat-plate BB. Manufacturer specification is 0.95 (+0.00, -0.05)).
If such disagreement takes place for flat-plate blackbody (metallic plate coated with black paint) for which emissivity measurement is not too difficult, one can imagine the situation with the cavity radiators.
It is impossible to calibrate radiometers, radiation thermometers, optical radiation detectors etc. without knowledge of radiation characteristics (primarily, effective emissivity) of blackbody calibration sources. Frequently, uncertainty in determination of effective emissivity is the predominant component in the uncertainty budget. Obviously, the best way to determine effective emissivity is measurements.
According to its definition, direct measurement of effective emissivity should be performed by comparison of radiation emitted by the blackbody source under consideration and radiation emitted by a perfect blackbody having the same temperature. However, the perfect blackbody is the physical abstraction that doesn't exist in the real world. There are two methods allowing to resolve this problem:
1) To perform absolute measurements of artificial blackbody radiation characteristics (e. g., spectral radiance in W·m-3·sr-1), assign a certain temperature to this radiator (real-world blackbody sources are always nonisothermal), then divide its measured quantity by homonymous quantity calculated for the perfect blackbody at the same temperature (e. g., using Planck's law for the case of spectral radiance).
2) To perform relative measurements by comparison of radiation emitted by the artificial blackbody under consideration and the better-quality artificial blackbody (such as fixed-point blackbodies).
Realization of both these methods is extremely complicated; it requires special instrumentation and highest qualification of operating personnel. Presently, such measurements can be conducted only in several world's leading metrological centers.
Another approach implies indirect measurement when cavity reflectance is measured instead of its emissivity. Indirect measurements can be implemented easily, for instance, using integrating sphere with the laser radiation source. However, this approach has serious disadvantages:
1) Usually, cavity reflectance is measured for the cavity temperature that significantly differs from the temperature at which the blackbody operates.
2) Often, it is difficult to provide geometry of cavity irradiation and collecting reflected radiation which correspond to geometrical conditions of collecting radiation emitted by the cavity when it is used for calibration. This violates the reciprocity principle and raises a query about correctness of effective emissivity determination.
3) Sometimes, such measurements are impossible at all, for example, due to the difference in dimensions of the blackbody and reflectometric device.
According to its definition, direct measurement of effective emissivity should be performed by comparison of radiation emitted by the blackbody source under consideration and radiation emitted by a perfect blackbody having the same temperature. However, the perfect blackbody is the physical abstraction that doesn't exist in the real world. There are two methods allowing to resolve this problem:
1) To perform absolute measurements of artificial blackbody radiation characteristics (e. g., spectral radiance in W·m-3·sr-1), assign a certain temperature to this radiator (real-world blackbody sources are always nonisothermal), then divide its measured quantity by homonymous quantity calculated for the perfect blackbody at the same temperature (e. g., using Planck's law for the case of spectral radiance).
2) To perform relative measurements by comparison of radiation emitted by the artificial blackbody under consideration and the better-quality artificial blackbody (such as fixed-point blackbodies).
Realization of both these methods is extremely complicated; it requires special instrumentation and highest qualification of operating personnel. Presently, such measurements can be conducted only in several world's leading metrological centers.
Another approach implies indirect measurement when cavity reflectance is measured instead of its emissivity. Indirect measurements can be implemented easily, for instance, using integrating sphere with the laser radiation source. However, this approach has serious disadvantages:
1) Usually, cavity reflectance is measured for the cavity temperature that significantly differs from the temperature at which the blackbody operates.
2) Often, it is difficult to provide geometry of cavity irradiation and collecting reflected radiation which correspond to geometrical conditions of collecting radiation emitted by the cavity when it is used for calibration. This violates the reciprocity principle and raises a query about correctness of effective emissivity determination.
3) Sometimes, such measurements are impossible at all, for example, due to the difference in dimensions of the blackbody and reflectometric device.
All serious researchers compute effective emissivities using Virial's software. The list of some publications concerned with the use of our old STEEP3 program can be found here. We invite our visitors to explore our programs of STEEP3 family (STEEP320, STEEP321, STEEP322, STEEP323) and INCA333 program for a cylindrical cavity with an inclined bottom in order to choose software the most suitable for solving your tasks. The Comparative Table will help you to make right choice.
Effective emissivity of a blackbody cavity depends to a great extent not only on values of emissivity (or reflectance) of cavity walls but also on angular distribution of radiation emitted or reflected by the internal surface of a cavity. Spectral and angular reflection properties of opaque material are described comprehensively by its spectral BRDF (bidirectional reflectance distribution function) introduced by F. E. Nicodemus et al. (1977). Usually, BRDF is considered as a function of five variables (wavelength, two angular coordinates for the incidence direction, and two for the viewing direction). Integration of the BRDF over hemispherical solid angle gives spectral directional-hemispherical reflectance (DHR). Spectral directional emissivity is equal to 1 minus the spectral DHR, if, according to the optical reciprocity principle, to substitute the direction of incidence for DHR by the direction of observation for directional emissivity. Measurement of BRDF for entire spectral and angular domains of interest is very laborious problem; usually BRDF is measured only for several wavelengths and several incidence angles; often, BRDF is measured only in the plane of incidence (so-called in-plane BRDF). Since the use of measured BRDF in calculations of radiation characteristics of blackbody cavities is problematic due to incompleteness of experimental data, the reflection models of various degree of adequacy and complexity are employed in radiometric calculations.
Diffuse Reflection Model
This is the simplest model which considers reflected radiation as perfectly diffuse (Lambertian), that is the radiance intensity of reflected radiation does not depend on the incidence angle and obeys Lambert's cosine law for the viewing angle; the radiance of reflected radiation does not depend on the incidence nor the viewing direction. Diffuse model of reflection allows to write integral equations for effective emissivity in a closed form, therefore diffuse model was used in early works on radiation properties of cavity despite its limited applicability to real-world materials.
Uniform Specular-Diffuse (USD) Reflection Model
The uniform specular-diffuse (USD) model is in common use in radiation heat transfer since 1960ths. USD represents reflected radiation as a sum of perfectly diffuse and perfectly specular components; both components do not depend on the incidence angle. This model can completely determined by the DHR and the diffusity D. The USD model is well-suited for Monte Carlo modeling of radiation heat transfer, although it might be too crude in some special cases. The special case of wavelength-independent reflectance is referred to as the Gray Uniform Specular-Diffuse (GUSD) model.
The inherent drawbacks of the USD model are (i) impossibility to fit it to measured data because the BRDF of the perfectly specular component is the Dirac delta-function that has infinite value in the direction of specular reflection, (ii) an ambiguity in separation of measured DHR onto diffuse and specular parts, and (iii) ignoration of reflectance dependence on the incidence angle.
General Specular-Diffuse (GSD) Reflection Model
This modification of specular diffuse model takes into account the dependence of the specular component on the incidence angle (usually, according to Fresnel's equation). The GSD model is more adequately reproduces angular behavior of material's DHR than the USD model, however, drawbacks (i) and (ii) of the USD model remain in force for GSD model.
Three-Component (3C) BRDF Model
The 3C BRDF model represents BRDF of a material as a weighted sum of three independent components: diffuse (Lambertian), glossy, and specular (actually, quasi-specular, i.e., having a specular lobe of small but finite width). 3C BRDF model has 8 parameters which can be fitted to the measured in-plane BRDFs. By supposing that proportions of components remain the same within a spectral range, one can separate the spectral DHRs of a material measured at one incidence angle onto diffuse, glossy, and specular terms. Read more...
This is the simplest model which considers reflected radiation as perfectly diffuse (Lambertian), that is the radiance intensity of reflected radiation does not depend on the incidence angle and obeys Lambert's cosine law for the viewing angle; the radiance of reflected radiation does not depend on the incidence nor the viewing direction. Diffuse model of reflection allows to write integral equations for effective emissivity in a closed form, therefore diffuse model was used in early works on radiation properties of cavity despite its limited applicability to real-world materials.
Uniform Specular-Diffuse (USD) Reflection Model
The uniform specular-diffuse (USD) model is in common use in radiation heat transfer since 1960ths. USD represents reflected radiation as a sum of perfectly diffuse and perfectly specular components; both components do not depend on the incidence angle. This model can completely determined by the DHR and the diffusity D. The USD model is well-suited for Monte Carlo modeling of radiation heat transfer, although it might be too crude in some special cases. The special case of wavelength-independent reflectance is referred to as the Gray Uniform Specular-Diffuse (GUSD) model.
The inherent drawbacks of the USD model are (i) impossibility to fit it to measured data because the BRDF of the perfectly specular component is the Dirac delta-function that has infinite value in the direction of specular reflection, (ii) an ambiguity in separation of measured DHR onto diffuse and specular parts, and (iii) ignoration of reflectance dependence on the incidence angle.
General Specular-Diffuse (GSD) Reflection Model
This modification of specular diffuse model takes into account the dependence of the specular component on the incidence angle (usually, according to Fresnel's equation). The GSD model is more adequately reproduces angular behavior of material's DHR than the USD model, however, drawbacks (i) and (ii) of the USD model remain in force for GSD model.
Three-Component (3C) BRDF Model
The 3C BRDF model represents BRDF of a material as a weighted sum of three independent components: diffuse (Lambertian), glossy, and specular (actually, quasi-specular, i.e., having a specular lobe of small but finite width). 3C BRDF model has 8 parameters which can be fitted to the measured in-plane BRDFs. By supposing that proportions of components remain the same within a spectral range, one can separate the spectral DHRs of a material measured at one incidence angle onto diffuse, glossy, and specular terms. Read more...
Virial International offers blackbody emissivity modeling programs of various levels of complexity (defining by the optical properties model adopted) and areas of applicability. STEEP320 and STEEP321 employ USD model of reflection; STEEP322 - GSD model; STEEP323 and INCA333 - 3C BRDF model as well as tools for fitting parameters of 3C model to the measured BRDFs and for separation of the spectral DHR components.
Cavity of various geometrical shapes are used as blackbody radiators. The choice of a cavity shape is determined by reflection properties of cavity wall material, conditions of cavity radiation collecting, and convenience of cavity manufacturing and treatment of its internal surface. The most frequently, cavities with axial symmetry are employed. All programs of STEEP3 family allow to model cavities formed by rotation of a non-self-intersecting polygonal line around an axis. If polygonal line has a lot of segments it can reproduce complicated surfaces of revolutions. Pictures below show some cavity shapes which can be modeled using STEEP320, STEEP321, STEEP322, and STEEP323. Figures in parentheses denote the numer of segments in the cavity generatrix.
Cone with diaphragm (2) Cylinder with diaphragm (3)
Cylindro-cone with diaphragm (3) Cone-cylindro-cone (3)
Cylinder-inner-cone with diaphragm (3) Cavity with reflector (9)
Cylinder with V-grooved bottom and diaphragm (23) Sphere (399)
Among non-axisymmetrical cavities, the cylindrical cavity with an inclined bottom is the most frequently used due to simplicity of fabrication and treatment. INCA333 allows to model just this type of cavities. Pictures below show two orthogonal cross-sections of a cylindrical cavity with an inclined bottom and flat annular diaphragm.
Effective emissivities and radiance temperatures derived from them depend on viewing conditions (i.e., geometrical conditions of collecting the radiation by a measurement device). Picture below shows the most common viewing conditions used in optical radiometry, radiation thermometry, and adjacent areas.
Normal viewing conditions might be considered as a special case of directional viewing conditions. Both these types correspond to the collimated beams. Such viewing conditions are approximately implemented for the radiometric or pyrometric measurement scheme that uses very long focal-length optics. If viewing beam radius is equal to zero (hypothetical case of infinitely thin viewing beam), distribution of the normal effective emissivity across the aperture of a cavity characterizes the spatial nonuniformity of the radiance emitted by a cavity; dependence of the directional effective emissivity upon the angle of entry of a viewing beam into cavity characterizes the angular nonuniformity of the radiance emerging by the given point of cavity aperture. For right conical viewing conditions, the beam axis is parallel the cavity axis. Integrated viewing conditions reproduce collecting of radiation emerging from a cavity by a circular detector without optical system. Hemispherical viewing conditions characterize total radiation losses of a cavity and can be considered as a special case of the integrated viewing conditions when the detector radius is equal to the cavity aperture radius and the distance between cavity aperture and detector is equal to zero.
Normal Directional
Right conical (convergent viewing beam) Right conical (divergent viewing beam)
Oblique conical (convergent viewing beam) Oblique conical (divergent viewing beam)
Integrated (detector is shown in blue) Hemispherical
The temperature distribution over the radiation surface of a blackbody cavity is the crucial factor together with the optical properties of cavity walls determining the effective emissivity and, consequently, the radiance temperature of a blackbody. High-quality blackbody radiator must be almost isothermal. Fixed-point and heat-pipe blackbodies as well as cavities immersed into thermostated liquid have the highest degree of temparature uniformity. However, real-world cavities are always nonisothermal. Even if the external surface of a cavity has ideally uniform temperature, the finite thermal resistance of a cavity wall (or layers forming it) and different radiative heat losses from the aperture for different parts of the cavity internal surface unavoidably lead to nonisothermality of radiating surface. Evaluation of this effect is possible currently only by computational methods.
Temperature nonuniformity of radiating surface affects significantly on effective emissivity of a blackbody, escpecially if the area within FOV (field-of-view) of the detector, radiometer, or radiation thermometer is nonisothermal. Nonuniformity of radiating surface leads to different contributions of radiation emitted by different elements of cavity walls at the same wavelengths to the resulting cavity radiation. This, in turn, makes the cavity spectrum non-Planckian. Effective emissivity is the ratio of cavity's radiation characteristic (e.g., spectral radiance) to the analogous characteristic of a perfect blackbody at the same wavelength and the same temperature. To avoid ambiguousness in determination of temperature for nonisothermal cavities, the reference temperature Tref is used in definition of effective emissivity. Depending of Tref choice, effective emissivity of a nonisothermal cavity might be less or greater than that of an isothermal cavity, and even be greater than unity. It is recommended to choose Tref between minimal and maximal temperature of cavity's radiation surface. Very often, the temperature of cavity bottom center is a good choice for Tref. Note, that the choice of Tref do affects only effective emissivity (it is fictitious quantity in a certain sense) but does not affect on measurable quantities (e.g., radiance, radiance temperature, radiation flux etc.)
Temperature nonuniformity of radiating surface affects significantly on effective emissivity of a blackbody, escpecially if the area within FOV (field-of-view) of the detector, radiometer, or radiation thermometer is nonisothermal. Nonuniformity of radiating surface leads to different contributions of radiation emitted by different elements of cavity walls at the same wavelengths to the resulting cavity radiation. This, in turn, makes the cavity spectrum non-Planckian. Effective emissivity is the ratio of cavity's radiation characteristic (e.g., spectral radiance) to the analogous characteristic of a perfect blackbody at the same wavelength and the same temperature. To avoid ambiguousness in determination of temperature for nonisothermal cavities, the reference temperature Tref is used in definition of effective emissivity. Depending of Tref choice, effective emissivity of a nonisothermal cavity might be less or greater than that of an isothermal cavity, and even be greater than unity. It is recommended to choose Tref between minimal and maximal temperature of cavity's radiation surface. Very often, the temperature of cavity bottom center is a good choice for Tref. Note, that the choice of Tref do affects only effective emissivity (it is fictitious quantity in a certain sense) but does not affect on measurable quantities (e.g., radiance, radiance temperature, radiation flux etc.)
Temperature nonuniformity of the area of cavity's internal surface within FOV (that is directly "viewed" by measuring device) results in drastic changes of spectral effective emissivity. Right graph shows the spectral effective emissivity plotted vs. wavelength for the same cavity as in previous example. Tref is also equal to 1300 K for two temperature distributions:
Temperature distribution 1. Temperature linearly decreases from 1300 K at the bottom center down to 1290 K at its periphey, then decreases linearly along cylindrical wall down to 1287 K; cavity diaphragm has constant temperature of 1287 K.
Temperature distribution 2. Temperature linearly increases from 1300 K at the bottom center up to 1310 K at its periphey, then increases linearly along cylindrical wall up to 1313 K; cavity diaphragm has constant temperature of 1313 K.
Temperature distribution 1. Temperature linearly decreases from 1300 K at the bottom center down to 1290 K at its periphey, then decreases linearly along cylindrical wall down to 1287 K; cavity diaphragm has constant temperature of 1287 K.
Temperature distribution 2. Temperature linearly increases from 1300 K at the bottom center up to 1310 K at its periphey, then increases linearly along cylindrical wall up to 1313 K; cavity diaphragm has constant temperature of 1313 K.
Temperature of isothermal cavities doesn't affect effective emissivity only if cavity's surround is the non-emitting media (like vacuum at 0 K) of if influence of background is negligible. Since the actual characteristics of background radiation is very seldom known, usually, one can consider cavity's surround as a perfect blackbody radiator with the "background temperature" Tbg.
The influence of Tbg = 300 K on the spectral normal effective emissivity of the above-mentioned cavity having isothermal walls at T = Tref = 400...1000 K is shown at left. Black line corresponds to the case of the isothermal cavity at Tbg = 0 K.
The radiance temperature measured in conditions of non-zero background will differ from the termodynamic temperature of a cavity not only due to non-unity effective emissivity but also due to contribution of the background radiation partially reflected by the cavity.
The influence of Tbg = 300 K on the spectral normal effective emissivity of the above-mentioned cavity having isothermal walls at T = Tref = 400...1000 K is shown at left. Black line corresponds to the case of the isothermal cavity at Tbg = 0 K.
The radiance temperature measured in conditions of non-zero background will differ from the termodynamic temperature of a cavity not only due to non-unity effective emissivity but also due to contribution of the background radiation partially reflected by the cavity.
For nonisothermal cavities, both effects (temperature nonuniformity of a cavity and radiating background) act conjointly. Besides, temperature effects cannot be considered in isolation from the angular distribution of radiation reflected by the cavity walls. In terms of the Monte Carlo ray tracing, this can be explained by the following fact: a ray undergoing multiple reflections from the cavity walls "accumulates" spectral radiances of all reflection ponts at their temperatures; for different angular distributions of reflected radiation, these points will be distributed alse differently.
All programs of STEEP3 family allow modeling of nonisothermal blackbodies with axially-symmetric temperature distributions. Temperatures can be defined over the discrete point set on the cavity generatrix; linear interpolation is used during ray tracing.
INCA333 allows to model only one-dimensional temperature distributions, along the axis of the cavity cylindrical part; the diaphragm is supposed to be isothermal at the temperature of the adjacent edge of the cylindrical part.
All programs of STEEP3 family allow modeling of nonisothermal blackbodies with axially-symmetric temperature distributions. Temperatures can be defined over the discrete point set on the cavity generatrix; linear interpolation is used during ray tracing.
INCA333 allows to model only one-dimensional temperature distributions, along the axis of the cavity cylindrical part; the diaphragm is supposed to be isothermal at the temperature of the adjacent edge of the cylindrical part.
In exemplification of this phenomenon, the dependences of the normal spectral effective emissivity on wavelength for gray diffuse cylindrical cavity with diaphragm having wall emissivity of 0.9 are shown below. Cavty bottom is isothermal at 1300 K; three temperature distributions are considered: (Isothermal) all cavity walls have uniform temperature of 1300 K; (10% Decrease) linear decrease of temperature along cylindrical surface from 1300 K down to 1287 K, diaphragm has uniform temperature of 1287 K; (10% Increase) linear increase of temperature along cylindrical surface from 1300 K up to 1313 K, diaphragm has uniform temperature of 1313 K. In all cases, Tref = 1300 K.
Precision of the Monte Carlo calculations is in inverse proportion to the square root of a number of rays traced; usually, it is enough to trace from 105 to 107 rays to achieve the accuracy of 10-4…10-6 in effective emissivity.
The logarithmic plot at right illustrates the stochastic convergence of the Monte Carlo modeling for several calculation processes peformed for a diffuse spherical cavity, for which the effective emissivity can be calculated exactly, using well-known analytical formula.
Monte Carlo calculation can be considered as a numerical experiment in some a measure similar to the ordinary natural experiment. Therefore, its accuracy can be assessed by performing repeated calculations and consecutive statistical processing of numerical experiment results.
Adequacy of the Monte Carlo modeling depends equally to a great extent upon two factors. The first is the temperature distribution over the blackbody walls, especially over their directly viewable areas. One can evaluate the effect of temperature non-uniformity on the radiation characteristics of a blackbody by performing calculations with several feasible temperature distributions.
The logarithmic plot at right illustrates the stochastic convergence of the Monte Carlo modeling for several calculation processes peformed for a diffuse spherical cavity, for which the effective emissivity can be calculated exactly, using well-known analytical formula.
Monte Carlo calculation can be considered as a numerical experiment in some a measure similar to the ordinary natural experiment. Therefore, its accuracy can be assessed by performing repeated calculations and consecutive statistical processing of numerical experiment results.
Adequacy of the Monte Carlo modeling depends equally to a great extent upon two factors. The first is the temperature distribution over the blackbody walls, especially over their directly viewable areas. One can evaluate the effect of temperature non-uniformity on the radiation characteristics of a blackbody by performing calculations with several feasible temperature distributions.
Direct measurements of effective emissivity of blackbody radiation sources are often extremely difficult or even impossible. Sometimes, computational methods are the only way to determine effective emissivity. Moreover, calculation of effective emissivities should be done at the design stage. Many computational methods for effective emissivities have been developed since the second half of XX century (see their overview here). They are based on the various physical and mathematical assumptions, have different areas of applicability and provide different degrees of accuracy. At present, the most comprehensive and flexible method for calculating radiation characteristics of blackbody radiators is the Monte Carlo method. It is based on the ray tracing algorithm that models radiation heat exchange among cavity walls and propagation of radiation from a blackbody to a radiation detector. The Monte Carlo method uses the ray (geometrical) optics approximation; such phenomena as polarization and diffraction are not considered.
The computational Monte Carlo algorithm employed in Virials's blackbody emissivity modeling software is based on the optical reciprocity principle and the technique of backward ray tracing. The history of a ray directed from the point of observation into a cavity is being traced until it leaves the cavity after reflections from the walls, or until its energy becomes less then the given value (flux threshold). The last point of reflection is considered as a birth point of a ray propagating in opposite direction. The spectral radiance acquired by a ray consists of the spectral radiance of own thermal radiation (computed by Planck’s law) in the point of last reflection in the viewing direction and the sum of spectral radiances in the points of successive reflectance computed along the ray trajectory and impaired by multiple reflections. Averaging of spectral radiances of a large number of rays traced allows to estimate the effective emissivity and radiance temperature of a blackbody.
Second factor crucially affected effective emissivity is the adequacy of angular dependences of the model adopted for optical properties of materials forming a blackbody cavity. The simplest (but very often insufficient) model is the diffuse model of reflection. The most powerful models must take into account bidirectional reflectance distribution function (BRDF) that describes angular distributions of reflected radiation for every direction of incident radiation. Since the measurement data required for such a model is very often incomplete or absent at all, the specular-diffuse model of reflection representing BRDF as a sum of the Lambertian (diffuse) and the perfect specular components was in common use up to date. However, BRDF for specular-diffuse model of reflection includes Dirac’s delta-function in the expression for specular component and cannot be fitted to measured BRDFs. Separation of reflectance into the perfect diffuse (Lambertian) and perfect specular components cannot be done unambiguously. Besides, specular components of reflection for real-world materials of blackbody cavities have small but finite divergence what can lead to effects which cannot be predicted within framework of specular-diffuse model of reflection.
Recently, we developed so-called Three-Component (3C) BRDF model (click here for detail) as well as software tools for fitting model’s parameters to in-plane BRDFs (i.e., BRDFs measured in the plane of incidence) and for separation of the spectral directional-hemispherical reflectance (spectral DHR) into three components - perfectly diffuse (Lambertian, or simply diffuse), glossy (having wide specular lobe), and quasi-specular (having narrow specular lobe; simply specular, for brief). For the moment, the 3C BRDF model is implemented in STEEP323 and INCA333 blackbody emissivity modeling programs.
Recently, we developed so-called Three-Component (3C) BRDF model (click here for detail) as well as software tools for fitting model’s parameters to in-plane BRDFs (i.e., BRDFs measured in the plane of incidence) and for separation of the spectral directional-hemispherical reflectance (spectral DHR) into three components - perfectly diffuse (Lambertian, or simply diffuse), glossy (having wide specular lobe), and quasi-specular (having narrow specular lobe; simply specular, for brief). For the moment, the 3C BRDF model is implemented in STEEP323 and INCA333 blackbody emissivity modeling programs.
