High-temperature measurement techniques for the application in photometry, radiometry and thermometry

Physics Reports 469 (2009) 205–269

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High-temperature measurement techniques for the application in photometry, radiometry and thermometry Jürgen Hartmann Physikalisch-Technische Bundesanstalt Braunschweig und Berlin, Abbestraße 2-12, D-10587 Berlin, Germany

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Article history: Accepted 11 September 2008 Available online 20 September 2008 editor: G. I. Stegeman PACS: 06.20 fb 07.20ka 07.60Pq 42.72 Bj 44.40.+a 95.75.fg

a b s t r a c t Well characterised sources of thermal radiation are essential for photometry, radiometry, and thermometry. They serve as reference radiators for the calibration of detectors and radiance sources. Thermal radiation sources are advantageous for this purpose compared to other radiance sources such as lamps or LEDs because they possess a continuous spectrum of the emitted spectral radiance, which, for blackbody sources, can be calculated analytically using Planck’s law of radiation. For application in thermometry, blackbody sources starting from temperatures near absolute zero to temperatures up to 3000 ◦ C are needed for the calibration of radiation thermometers. For application in photometry and radiometry high intensity sources of radiation in the visible and UV region of the optical spectrum were required. This latter requirement is met by blackbody sources at temperatures well above 2000 ◦ C. An ideal reference source should always emit the same amount of radiation at any time of use. This is realised by fixed-point radiators. Such radiators are based on a phase transition of a substance, at high temperatures the melting and freezing points of metals. However, current metal fixed-points are limited to relatively low temperatures. In the present work innovative techniques necessary for research into high-temperature thermal radiation sources are developed and thoroughly described. Starting with variable temperature blackbody sources the techniques required are: Precise apertures determination and detailed characterisation of the applied optical detectors. The described techniques are then used to undertake research into the development of high-temperature fixed-points above the copper fixed-point for application in photometry, radiometry, and thermometry. Applying these sophisticated techniques it was shown that these new high-temperature fixed-points are reproducible and repeatable to better than 100 mK at temperatures up to nearly 3200 K. Finally, a forward look is given that shows the potential of such fixed-points for improving traceability and accuracy in photometry, radiometry, and thermometry. This work forms the foundation for accurate and practical applications of high-temperature fixed-point sources and sets the international benchmark for measurement techniques in photometry, radiometry, and thermometry. This work will open the use of the novel high-temperature fixed-points in an improved International Temperature Scale and will significantly improve and ease the realisation and dissemination of the SI base unit candela, significantly reducing the uncertainty for industrial measurements in these fields. © 2008 Elsevier B.V. All rights reserved.

Contents 1.

Introduction............................................................................................................................................................................................. 206 1.1. History of thermal radiation ...................................................................................................................................................... 207

E-mail address: [email protected]. 0370-1573/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physrep.2008.09.001

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1.2. History of the definition of the candela..................................................................................................................................... 213 1.3. Thermodynamic temperature, the kelvin and the International Temperature Scale of 1990 ............................................... 215 1.4. Geometry of a radiometric measurement................................................................................................................................. 216 Technical realisation of blackbody radiation ........................................................................................................................................ 218 2.1. Blackbody emissivity .................................................................................................................................................................. 219 2.2. Consideration of the effective refractive index of blackbody radiation .................................................................................. 220 2.3. The blackbody radiators ............................................................................................................................................................. 222 2.3.1. Low temperature blackbodies .................................................................................................................................... 223 2.3.2. High-temperature blackbody ..................................................................................................................................... 224 2.3.3. High-temperature furnace with carbon cavity .......................................................................................................... 224 2.3.4. High-temperature furnace with pyrolytic graphite cavity ....................................................................................... 225 Radiometric precision apertures ............................................................................................................................................................ 230 3.1. Absolute determination of the effective optical area of the apertures ................................................................................... 231 3.2. Relative determination of the area of the precise apertures ................................................................................................... 234 Spectral responsivity of detectors used in this study ........................................................................................................................... 238 4.1. Optical power measurement with the radiation thermometry cryogenic radiometer (RTCR) ............................................. 238 4.2. Characterisation of transfer detectors ....................................................................................................................................... 238 4.2.1. Non-linearity................................................................................................................................................................ 239 4.2.2. Spatial uniformity ........................................................................................................................................................ 239 4.2.3. Temperature coefficient .............................................................................................................................................. 240 4.2.4. Long term stability....................................................................................................................................................... 240 4.2.5. Spectral responsivity uncertainty at distinct laser lines ........................................................................................... 240 4.2.6. Model of quantum efficiency and spectral responsivity scale.................................................................................. 242 4.3. Model for the temperature dependence of the spectral responsivity..................................................................................... 243 Transfer of the spectral responsivity of the reference detector to the filter radiometer ................................................................... 248 5.1. Experimental set-up ................................................................................................................................................................... 249 5.2. Temperature dependence of the FRs ......................................................................................................................................... 250 5.3. Non-linearity of the spectral responsivity ................................................................................................................................ 251 5.4. Polarisation effects...................................................................................................................................................................... 251 5.5. Stray light .................................................................................................................................................................................... 251 5.6. Uncertainty of the filter radiometer calibration ....................................................................................................................... 252 Development of new high-temperature fixed-points above 1400 K................................................................................................... 252 6.1. Metal-carbon eutectics ............................................................................................................................................................... 252 6.2. Fixed-points made from metal (carbide) carbon eutectics ...................................................................................................... 255 6.3. Investigation and temperature determination of metal(carbide) carbon eutectics............................................................... 256 Application of the new high-temperature fixed-points in photometry, radiometry and thermometry .......................................... 261 7.1. Photometry and radiometry ...................................................................................................................................................... 261 7.1.1. Standard sources for thermal radiation ..................................................................................................................... 261 7.1.2. Check and calibration of the spectral responsivity of photometers and radiometers ............................................ 264 7.2. Thermometry .............................................................................................................................................................................. 265 Summary ................................................................................................................................................................................................. 266 Acknowledgements................................................................................................................................................................................. 266 References................................................................................................................................................................................................ 267

1. Introduction Light, i.e. optical electromagnetic radiation within a wavelength range between 380 and 780 nm is most important for human beings and an enormous effort has been spent developing reliable and stable radiation sources to investigate light detectors and sources. Accurate measurement of light is essential for developing new and economic light sources for private and automotive use. Especially the adoption of the modern solid state light sources, like LEDs, is requiring an accurate spectral measurement of the emitted radiation. Additionally, energy efficient lighting and heating of buildings is strongly depending on the online measurement on the incoming sunlight. One actual problem, widely discussed in the scientific and public community relying on highly stable radiation sources is global warming. The Earth’s surface temperature has been monitored for a long time and the results of this are shown in Fig. 1 [3]. Fig. 1 shows an increase of the average Earth surface temperatures of about 0.8 K over the last 150 years. These measurements can only be trusted, if the temperatures measured in 1860 can be directly compared with those measured in 2004. This requires the presence of methods how to determine temperature, which is valid over these time scales. This was done by implementing an International Temperature Scale, in detail described in Section 1.3. The temperature increase observed in Fig. 1 is attributed to anthropogenic factors, especially the production of CO2 due to the use up of fossil fuels [2]. Thus, temperature rise is due to the positive feedback the ecosystem has on the greenhouse effect driven by the energy delivered to the Earth by the Sun [2]. The irradiance of the Sun at the Earth’s outer atmosphere is about 1300 W/m2 . This

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Fig. 1. Difference of the mean Earth surface temperature from the average temperature between 1961 and 1990 [1].

Fig. 2. Solar irradiance at the earth atmosphere as a function of time [3] (courtesy Nature Publishing Group).

irradiance is not stable in time, but varies with a periodicity of about 11 years, the time for the so-called Sun-cycle as shown in Fig. 2 [3]. Fig. 2 reveals a significant variation of the solar irradiance at the Earth’s atmosphere, which significantly changes the global energy budget. For example, it is thought that the drop in the solar irradiance between 1982 and 1986 completely compensated for the increase in Earth surface temperature due to the anthropogenic effects (Fig. 1) [3]. Therefore, monitoring the radiative output of the Sun on a 0.1% uncertainty level is essential for accurate monitoring and prediction of global warming and to give politicians the right information for their decisions. Besides this global climate problem, high-temperature measurements are of great concern for the lightning industry, because the reliability of tungsten filament lamps is strongly related to the absolute temperature of the tungsten material. However, as tungsten is not a blackbody but has an emissivity lower than one and dependent on wavelength, absolute temperature measurement of the tungsten material requires an accurate high-temperature reference source. The structure and change in structure of tungsten is strongly correlated to the maximum temperature during use as can be seen in Fig. 3 [4]. The structure and size of the grains changes with temperature, changing the resistance and the emissivity of the tungsten wire. It is therefore essential to measure the absolute temperature of the tungsten filament with high accuracy. As neither the emissivity of the tungsten nor the transmission of the glass bulb is known, accurate absolute high temperature radiation sources are needed to determine the absolute temperature of the tungsten strip or filament. 1.1. History of thermal radiation Thermal radiation, generated and emitted by every hot body, has been the first source of artificial light used for lightning when the Sun’s light is not available. The origin of thermal radiation is the thermal movement of the charged elementary modules matter is built of. The oppositely charged particles moving with respect to each other are forming elementary antennas, so-called dipoles. Such moving dipoles emit electromagnetic radiation, which is called thermal radiation. At high temperatures this emitted thermal radiation can be seen by the human eye as emitted light. In the beginning of the industrial revolution at the end of the 19th century, besides candles, gaseous light was the only reliable source of optical radiation. However, the spectral radiance or luminance of such a source could not be calculated

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Fig. 3. Crystallite grains of tungsten wire at low (a) and high (b) temperatures [4].

analytically. To produce a reliable source of optical radiation, as it was done by the Hefner candle, a detailed and elaborated description was required. Already in the mid of the 19th century, basing on thermodynamic considerations, Gustav Kirchhoff proposed the existence of an absolute radiation source [5]. The radiation emitted by such an absolute radiation source should not depend on the optical properties of the surface of the source or the material used, but only on its temperature. Additionally, Kirchhoffs’ investigation led him to postulate that the ability of a body to emit radiation, its emissivity, is equal to its ability to absorb radiation, its absorptivity. A perfect thermal radiator should, therefore, also be a perfect absorber and is therefore called a blackbody, because it absorbs all incoming radiation and consequently appears black. Kirchhoff restricted himself to theoretical considerations, not seeking a practical realisation of such a blackbody. But he argues that it was scientifically and technologically important to find the analytical representation of the spectral radiance emitted by such a blackbody. As a result of his theoretical considerations he was convinced that such an analytical function describing blackbody radiation must be quite simple in structure. Kirchoffs’ theoretical investigations inspired a variety of famous scientists, like Willy Wien, Ferdinand Kurlbaum, Max Planck and others, to work in the field of blackbody radiation [6]. At the same time, i.e. in the last twenty years of the 19th century the revolution of electricity in nearly every area of public life occurs and electrical lightning starts to replace gaseous lightning at this time. The famous industrialist Werner von Siemens sought a technical standard source of optical radiation, to help decide whether the traditional gaseous or his modern electrical street-lightning was superior. To initiate the development of such a standard radiation source – and other technological relevant measurement techniques – he decided to fund the foundation of an independent governmental measurement institution. Mainly due to the efforts of Werner von Siemens and Herman von Helmholtz, who later becomes its first president, the Physikalisch-Technische Reichsanstalt (PTR) was erected in 1887 on an area donated by Werner von Siemens. (In 1953 the PTR was renamed in Physikalisch-Technische Bundesanstalt (PTB)). One of the first tasks of the PTR was the investigation of blackbody radiation. This required that a technical solution was found for Kirchhoffs’ theoretically predicted blackbody sources. Willy Wien and Ferdinand Kurlbaum were the first who used cavity radiators as technical approximations for blackbody sources [7]. Such cavity radiators were formed by a cavity heated by hot fluids or gases of the desired temperature. These were then used at the end of the 19th century at the PTR for fundamental experiments in the temperature range from 90 and 1000 K. Fig. 4 shows a picture of the radiometry laboratory of the PTR around the year 1900. The intention of Willy Wien was to verify experimentally his law of blackbody radiation, which he had already discovered in 1896 [9]. And indeed, the first series of experiments for temperatures up to about 1000 K demonstrate close agreement between his theory and experiment [10]. Typical blackbodies for realising such temperatures between 90 K and 1000 K are shown in Fig. 5 [11]. For the high temperatures up to about 1000 K well stirred liquid saltpetre was used (left side of Fig. 5), while for the lowest temperatures down to 90 K liquid nitrogen was used as cooling agent (right side of Fig. 5). For investigation of thermal radiation at even higher temperatures up to 1800 K a new type of blackbody was developed at the PTR, the so-called electrically heated blackbody [12]. In this blackbody the radiating cavity was built of a thin platinum layer heated by direct electrical current (Fig. 6). For the highest temperatures a blackbody with the schematics shown in Fig. 7 was constructed [13]. For this kind of blackbody a graphite tube was directly heated by an electrical current – a concept still in use today – up to temperatures of about 2600 K. At the end of the 19th century quite advanced detectors of thermal radiation had been constructed. These were bolometers, made from a thin (<1 µm) layer of platinum, one of the first micro-technological devices [14]. The detector element can be seen in Fig. 8.

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Fig. 4. Temperature Radiometry laboratory of the PTR around the year 1900 [8].

Fig. 5. Typical blackbodies for realising temperatures down to 90 K (right) up to 1000 K (left) [11].

Fig. 6. Picture of the electrically heated blackbody [12].

The whole detector is formed out of two such elements (see Fig. 9), where one detects the thermal radiation from the source and the other is to allow for the compensation of environmental temperature changes.

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Fig. 7. Electrically heated graphite blackbody for temperatures up to 2600 K [13].

Fig. 8. Bolometer detector element [14].

Fig. 9. Complete area bolometer with sensing element and compensation for environmental temperature changes (left) [14] and line bolometer (right) [15].

Applying these new types of blackbody radiators revealed significant differences between Wien’s theory and the experimental data especially at high temperatures in the long wavelength range [16]. The results of these experiments are shown in Fig. 10.

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Fig. 10. Comparison of experimental with the theoretical predicted data using Wien’s law of radiation [16].

It can be seen that Wien’s approach accurately specifies the spectral blackbody radiance for low temperatures and short wavelengths, but fails at longer wavelengths and higher temperatures. In this region another analytical representation, obtained by Rayleigh and Jeans [17], give excellent agreement with experiments, but unfortunately failed at low temperatures and short wavelengths, leading to an unphysical infinite spectral radiance at the shortest wavelengths, the socalled ultra-violet catastrophe. So by 1899 two representations for spectral radiance emitted by a blackbody were known, accurately describing the spectrum for short (Wien [18]) or long wavelengths (Rayleigh and Jeans [17]). It was the work of Max Planck, then Professor at the University of Berlin, to discover an empirical representation, which interpolates the former both solutions and fits exactly the experimental data for all wavelengths and all temperatures. However, Max Planck was not satisfied by this empirical function, but also sought a sound theoretical basis. In October 1900 he succeeded in doing so but only by making a revolutionary assumption. The assumption was that the energy of thermal radiation is not distributed continuously, but is formed of small but finite energy packets [19]. This means that a change in the energy of light or thermal radiation could not be performed continuously, but only in steps. This assumption was contradictory to the well known proposition of those days ‘‘natura non facit saltus’’, nature does not make steps. Planck’s fundamental assumption was marvellous in the history of physics being the beginning of quantum mechanics, which has been repeatedly proven and is now a well established pillar of modern physics. In detail, Wien’s formula, describing the energy emitted by a blackbody per unit area and wavelength interval in a certain direction, Lλ,s , is based on classical thermodynamic considerations. This is given in Eq. (1.1) [18] Lλ,s (λ, T ) =

c1

λ

5



exp −

c2 

λT

,

(1.1)

with c1 = 2hc 2 and c2 = hc /k (h Planck’s constant, k Boltzmann constant, c velocity of light). Eq. (1.1) is depicted in Fig. 11. The empirical formula developed again through classical thermodynamic considerations by Rayleigh and Jeans is [17] Lλ,s =

c1 λT . λ5 c2

(1.2)

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Fig. 11. Spectral radiance calculated according Planck’s, Wien’s and Rayleigh–Jeans equations.

Fig. 12. Graphical representation of Planck’s, Wien’s and Rayleigh–Jeans formula of thermal radiation. The graph shows the second derivative of the entropy with respect to internal energy plotted versus internal energy.

Fig. 11 shows Wiens and Rayleigh–Jeans functions of Lλ,s as a function of wavelength, clearly showing the significant disagreement in the short wavelength region. In this representation it seems to be highly unlikely that a simple formula could be found that could be able to interpolate both functions. In those days it was quite usual to describe physical procedures thermodynamically, using concepts like internal energy and entropy. Thus, plotting the second derivative of the entropy of these functions with respect to their internal energy as a function of internal energy as shown in Fig. 12 an interpolation seems to be quite easy [20]. The formula developed by Planck is [20] Lλ,s =

c1

1

λ5 exp

c2 λT




−1

.

(1.3)

The interpolating character of the Planck formula can be seen by the Taylor evolution of the denominator given below in Eq. (1.4) exp

c  2

λT

=1+

c2

λT

−

1  c2 2 2

λT

± ···.

(1.4)

For λT much smaller than unity the 1 in the denominator of Eq. (1.3) can be neglected, yielding Wien’s formula. In the opposite extreme, i.e. λT much larger than unity the Taylor evaluation equation (1.4) of the exponential function can be truncated after the second term, resulting in the Rayleigh-Jean’s approximation. The validity of Planck’s function has been verified by numerous experiments often quite impressively, and offers the possibility for a calculable source of optical radiation applying blackbodies. In fact, Planck’s law of radiation was the beginning of quantum mechanics and opened the door for an impressive and unbelievable surge of physical and technological developments.

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Fig. 13. Schematic diagram of a fixed-point blackbody radiator (left) and temperature evolution of the fixed-point during a step change heating of the furnace (right).

1.2. History of the definition of the candela In the beginning of lighting technology every country had its own definition of luminous intensity, which was only rather poorly defined by flame standards [21]. In 1881 it was recommended to use the violle, the radiation temperature of a platinum surface during melting [22]. Unfortunately, the reproducibility of the violle was rather poor and flame standards still remained the best standards for luminous intensity. The most commonly used flame standard was the Hefner candle (HK). This standard used the luminous intensity perpendicular to the flame of the amylacetate flame described by HefnerAltneck. The dependence of the luminous intensity J of such a flame from the air humidity H, the air pressure p (in Torr), and the CO2 content of the air c(CO2 ) (l/m2 ) was well known (see Eq. (1.5)) as described in [21] J = 1 − 0.0055 · (H − 8.8) + 0.00015 · (p − 760) − 0.0072 · (c (CO2 ) − 0.75) HK.

(1.5)

In 1896 the bougie decimale, basing on the violle was internationally adopted as new base unit of the luminous intensity, however, Germany still remained on the Hefner candle. In 1909 an international agreement between the United Kingdom, France, and the United States of America adopted the International candle, based on carbon filament lamps, while Germany still remains at the Hefner Candle [21], although the International candle was approved in 1921 by the Commission International de l’Eclairage (CIE). Like the Hefner Candle also the carbon filament lamps have not been stable enough and such a definition could only be provisional. Although the application of blackbodies for the reliable and accurate generation of optical radiation was known since the late 1890’s, it took until 1933 for the luminous emission of a blackbody at the freezing temperature of platinum (∼2045 K) to be adopted as a new photometric unit. Following that development in 1937 the CIE and the CIPM prepared the definition of the new candle, announced by the CIPM in 1946. It was finally ratified in 1948 by the 9th CGPM, which adopted a new international name for this unit, the candela (symbol cd). Finally, in 1967 the 13th CGPM gave an amended version of the 1946 definition, namely The candela is the luminous intensity perpendicular to (1/6) × 10−5 m2 of the surface of a blackbody at a temperature of the melting platinum at a pressure of 101 325 m−1 kg s−2 [23]. As a consequence of these developments, the definition of the candela based on the Hefner Candle was valid until 1948 in Germany. Only then was blackbody radiation used for the definition of the candela. For a reliable source, yielding the same spectral radiance at any time, a blackbody radiator at the temperature of freezing platinum was used. During the melt and the freeze the temperature of the molten platinum – or any other metal – remains stable as long as not all material is molten or frozen. This type of blackbody is called a fixed-point blackbody, because it is used to realise the thermal radiation at a fixed-temperature, the melting and freezing temperature of the metal, at any time. In principle the radiating cavity of a fixed-point radiator is immersed in the metal as shown in Fig. 13. When a fixed-point is heated inside a furnace with a step like temperature profile, from below the melting temperature to above and vice versa, as shown in Fig. 13, two plateaux can be observed. These correspond to the melt and freeze of the pure metal. During these plateaux the temperature is stable within a very narrow temperature range and the fixed-point then serves as a stable and calculable source of thermal radiation. A practical set-up for realising the candela using a platinum blackbody is shown in Fig. 14 [24]. However, due to stability problems of the platinum fixed-point cells, which significantly effected the melting and freezing temperature of the platinum [25], the realisation of the candela was accurate to about 0.2% [22], see Fig. 15. This is several orders of magnitude less accurate than the other units of the SI are realised. Therefore, the definition of the candela was replaced by the 16th CGPM by a more academic definition in 1979, stating that one candela is . . . the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 × 1012 Hz and that has a radiant intensity in that direction of 1/683 W per steradian. This theoretical definition does not predict a practical scheme for the realisation of the candela. Nowadays the realisation of the SI-unit candela at the PTB is made by absolutely calibrated photometers and a network of tungsten strip lamps [26]. However, the obtained accuracy of the candela is still at the level of 0.1%, much less accurate than most other SI units [27].

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Fig. 14. Realisation of the SI unit candela according the definition from 1948 [24].

Fig. 15. Results of the international comparison of the candela in the last century. Graphic from M. Stock (BIPM, Paris).

Looking at the history of international comparisons of the candela as realised by several national metrology institutes shown in Fig. 15 it can clearly be seen that the redefinition of the candela did not significantly improve the international agreement of the realisation of the candela. To significantly reduce the uncertainty of the realisation of the candela improved high-temperature fixed-points of superior quality to that of the platinum fixed-point and improved high-temperature measurement techniques are required. Already in 1996 the CCT (Consultatif committee for thermometry of the Bureau International de Poids e Measure) set a flag that high-temperature fixed-points above 2300 K with reproducibility in temperature below 100 mK are highly needed [28]. The first promising attempt for manufacture of fixed-points fulfilling the recommendation T2 of the CCT from 1996 have been the metal carbon and metal-carbide carbon (M(C)–C)-based eutectic fixed-points introduced by Yoshiro Yamada from the National Metrology Institute of Japan in 1999 [29]. Since 1999 these fixed-points have been the topic of world wide investigations. In Europe these investigations have been lead through the FP5 EU project ‘‘Novel high-temperature metal carbon eutectic fixed-points for radiation thermometry, radiometry, and thermocouples’’ acronym HIMERT [30]. The aim of this international project was the detailed investigation of the M(C)–C eutectics for their practical application in radiometry and thermometry. For that purpose, advanced radiation thermometry and radiometry methologies have to be developed. To take full advantage of these fixed-points, as standard sources of thermal radiation, their thermodynamic melting and freezing temperatures must be determined with the lowest possible uncertainties (∼100 mK and lower). The present work describes the development of these improved methods necessary for the development and the investigation of such novel fixed-points. These measurement techniques have been developed at the PTB during the last

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years and include methods for accurately calibrating and well characterising detectors as well as sophisticated blackbody and furnace design. 1.3. Thermodynamic temperature, the kelvin and the International Temperature Scale of 1990 Temperature is probably the most important variable of state which influences almost every physical, chemical, and biological process. It determines the speed of chemical reactions, the reproduction of living cells, the efficiency of thermal engines, and the emission of thermal radiation. Everyday life also depends strongly on temperature. This ubiquity makes temperature one of the most – if not the most – frequently measured physical quantity [31]. The importance of temperature measurement for industrial processes was realized quite early and in 1927 the Conférence Générale des Poids et Measures (CGPM) decided to introduce an International Temperature Scale (ITS) to harmonize the world wide temperature measurements, and guarantee their comparability over time. In contrast to other base quantities of the International System of Units (SI) like length or mass, temperature is an intensive not extensive quantity. This means the temperature 2T can not be realised by joining two systems of temperature T . This is why a temperature scale has to be defined for the measurement of temperature. Using the freezing and the boiling point of water at standard atmospheric pressure (101 325 Pa) as reference temperatures, Anders Celsius defined a temperature scale in 1742 which is still in use today. By dividing this temperature interval by 100, the temperature unit degree Celsius (symbol ◦ C) was defined. In 1854, William Thomson, better known as Lord Kelvin, proposed a new temperature scale based on the second law of thermodynamics using the absolute zero temperature and one additional temperature fixed point. A hundred years later, in 1948, the CGPM decided to follow his proposal. As the temperature fixed point the triple point of water was chosen, i.e. the temperature at which the solid, liquid, and gaseous phases of water are in equilibrium. The temperature of the triple point of water (TPW) was fixed to be 273.16 K defining the temperature unit kelvin (symbol K) by 1 K = TTPW /273.16.

(1.6)

From this definition it follows that temperature differences are exactly the same irrespective of whether the kelvin or the Celsius temperature scale is used, while absolute temperatures differ by 273.15 K. Temperatures measured according to the Celsius temperature scale are denoted as t, while temperatures measured by the kelvin temperature scale are referred to as T . Conversion of temperatures measured in degrees Celsius into kelvin temperatures is made through T /K = t /◦ C + 273.15.

(1.7)

In principle, any fundamental relation incorporating temperature can be used for temperature measurement. A wellknown method is gas thermometry which relies on the ideal gas equation, that is the product of pressure p and volume V is directly proportional to the temperature T of the gas. Other, less commonly applied, primary thermometric methods are noise thermometry which uses the Nyquist noise equation relating thermal noise in an electrical resistor to its temperature, and acoustic thermometry relying on the relationship between the speed of sound and temperature of an ideal gas. More information on these techniques is given e.g. in [32]. A non-contact temperature measurement technique measures the emission of thermal radiation from a blackbody and relates it to the blackbody temperature by Planck’s law of radiation. All temperature measurement methods relying on such fundamental thermodynamic equations measure the absolute or thermodynamic temperature T. For practical everyday temperature measurements, primary temperature measurement techniques are too complex, time-consuming, and expensive. To meet the increasing demand for simplified but reliable and world wide comparable temperature measurements in science and industry the CGPM in 1927 adopted a detailed description of a temperature scale based on a number of reproducible temperatures, so-called fixed-points, to which numerical values of thermodynamic temperature were assigned, and well-defined standard instruments to be calibrated at these fixed points. The calibration of these standard instruments furnished the constants for the formulae defining temperatures in the various temperature ranges between the fixed-points. This detailed set of instructions was called the International Temperature Scale of 1927. Based on technological and scientific progress, the International Temperature Scale is revised from time to time so that the temperatures realised according to this scale are the closest possible practical approximation of the corresponding thermodynamic temperatures. At present the International Temperature Scale of 1990 (ITS-90) is valid [33]. The foundation of the ITS-90 is a set of temperature fixed-points with defined temperatures, which are embodied by phase transitions of pure materials and the triple point of water. These provide highly reproducible and stable temperatures, which can be used as reference temperatures for the calibration of temperature measurement devices. The thermodynamic temperatures of these phase transitions have been accurately measured by applying one of the abovementioned fundamental temperature measurement techniques [34]. Besides the temperature fixed-points, the ITS-90 also specifies the instruments required to establish the scale with low uncertainty between the fixed-points. In the most important practical temperature range from 13.8033 to 1234.93 K (freezing point of silver), the change in resistance of a platinum wire with temperature has to be used for interpolation between the fixed points. Above 1234.93 K, temperature measurements in ITS-90 are specified by measuring the spectral radiance ratio of the thermal radiation source with the unknown temperature to that of a blackbody operated at the temperature of freezing silver, gold or copper and then using

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Fig. 16. Schematic of the defining fixed points and interpolating instruments of the International Temperature Scale from 1990 above 0.01 ◦ C.

Planck’s law of radiation for evaluating the unknown temperature. A schematic of the defining fixed points and interpolating instruments of the ITS-90 is given in Fig. 16. Temperatures measured according to the ITS-90 are denoted by T90 (K) or t90 (◦ C), and are only approximations, though rather good ones, of the thermodynamic temperature T . Although any possible difference between T and T90 is very small, and at the moment not significant for most industrial and scientific applications, in the temperature range from 13.8033 to 1234.93 K the standard uncertainty of the ITS-90 with respect to the thermodynamic temperature extends to 40 mK at the freezing point of silver. For temperatures above the freezing point of silver, the uncertainty increases quite rapidly. This is because above this temperature, due to the lack of stable and reproducible fixed-points, the ITS-90 is based on extrapolation and the uncertainty in the realisation of the reference temperature of the silver, gold or copper fixed points with respect to the true thermodynamic temperature propagates according to [35]




u T90,therm =



T90 T90,ref

2

u T90,ref therm .




(1.8)

With the expanded uncertainty in the realisation of the thermodynamic temperature of the gold fixed-point of 100 mK then Eq. (1.8) gives the theoretical limit of the uncertainty measurement according the ITS-90 and is shown in Fig. 17. It can be seen that the base uncertainty limit of the ITS-90 expands from 100 mK at the gold fixed-point to 600 mK at 3250 K. This principle uncertainty limit is further increased by the additional uncertainty contribution introduced by the radiation thermometer necessary to perform the measurements. As a rule of thumb an expanded uncertainty of 0.2% for a typical highquality radiation thermometer will increase the uncertainty in temperature measurement to more than 1500 mK at high temperatures of 3250 K. This situation can be significantly improved if the non-contact high-temperature part of the ITS can be put on the same footing as the contact thermometer-based low temperature part of the ITS-90, i.e. based on interpolation instead of extrapolation. The novel high-temperature fixed points with well-known thermodynamic temperatures and a reproducibility and stability of the order of 100 mK will radically improve practical temperature measurements above the freezing point of copper. 1.4. Geometry of a radiometric measurement Due to the limited thermal stability of the materials used and their relatively low melting temperatures, high temperature measurements using contact techniques are generally limited to temperatures below 2500 K [36]. The temperature measurement methods of choice in the high-temperature range are radiation thermometry and absolute radiometry. Both

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Fig. 17. Theoretical (solid line) and practical limit when using a radiation thermometer with an expanded uncertainty of 0.2% (dotted line) of the uncertainty of temperatures measured according the International Temperature Scale from 1990.

Fig. 18. Geometry of a radiometry experiment.

of them are optical methods and the difference between these two methods is a formal one. While radiation thermometry relies on the fixed-points defined in the ITS-90, radiometric methods are absolute ones, independent from the defined temperatures of the ITS-90 fixed-points, but traceable to an absolute detector standard. In general the radiation thermometry or radiometry instrument consists of a detector and an optical system. The optical system is formed by apertures and optional imaging optics, which define the field of view of the instrument. Although imaging optics increase the field of view leading to higher signals, their main disadvantage is a possible contamination of the surfaces by dust and dirt, disturbing the ideal optical properties. For the purpose of this work radiometry devices without imaging optics have been used. However, the radiation thermometric devices used have been equipped with imaging optics. The principle geometry of a radiation thermometry and radiometry experiment is shown in Fig. 18. The main parts of the radiometry experiment shown in Fig. 18 are the radiation source, the apertures defining the geometry and the detector unit. The signal IPhoto of the filter radiometer unit shown in Fig. 18 is dependent on the spectral radiance Lλ , s (λ, T ) emitted by the blackbody, the geometrical factor G determined by the areas of the two apertures A1 and A2 and the distance d between them as well as the spectral responsivity s(λ) of the filter radiometer unit. The signal can be evaluated according the following equation. ∞

Z

Lλ,s (λ, T ) s (λ) dλ

IPhoto = G ·

(1.9)

0

with 2π r12

G= r12 + r22 + d2 +

q

r12 + r22 + d2


2

. − 4r12 r22

r1 and r2 are the radius of the source area A1 and the aperture area A2 shown in Fig. 18. The schematic drawing of the two radiation thermometers used within this study, the LP3 radiation thermometer from KE Technologie Stuttgart and the TSP radiation thermometer from VNIIOFI is shown in Fig. 19. The main difference between the measurement geometry shown in Fig. 18 is the used optics. Due to this imaging system such radiation thermometers measure radiance instead of irradiance. To make use of Eq. (1.9) and the analytical solution found by Max Planck, nearly ideal blackbody radiation has to be generated by the source. The technical solution for generating blackbody radiation is a cavity radiator, the principles of which

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Fig. 19. Schematic drawing of the radiation thermometers LP3 (left) [129] and TSP (right) [130].

Fig. 20. Enhancement in emissivity by geometrical effects according Eq. (2.1).

are described in Chapter II. The fabrication and characterisation of the precision apertures defining the optical geometry and therefore the parameter G of Eq. (1.9) are described in Section 3. The detection unit, the calibration of its spectral responsivity s(λ) with respect to the primary detector standard and its characteristics are presented in Section 4. The optimized devices of Sections 2–4 are used to determine the thermodynamic temperature of novel high-temperature fixed-points the results of which are shown in Section 5. In Section 6 application of the new fixed-points as reference standards in thermometry as well as in radiometry and photometry is given. 2. Technical realisation of blackbody radiation To make use of the analytical solution (Eq. (1.3)) found by Max Planck an ideal blackbody has to be used. Already Willy Wien and Ferdinand Kurlbaum in 1895 found that a cavity uniformly heated and having a small opening in comparison to its inner surface, will serve as a source of nearly ideal blackbody radiation [7]. The emitted radiation is nearly blackbody like, if, beside the small ratio of radiating area to inner surface area, the temperature is the same all over the inner surface of the cavity. The determination of the resulting emissivity is one of the key problems of radiation thermometry and radiometry. High emissivities close to one were difficult to measure and are, therefore, usually calculated using analytical or numerical methods. A simple method for estimating the emissivity εcavity of a cavity radiator with a wall emissivity εwall , length l and radius r was given by Bauer and Bischoff in 1970 with [37]

εcavity = 1 −

1 − εwall

εwall

·

1 1+

l2 r2

.

(2.1)

Eq. (2.1) is a simple representation describing the enhancement of emissivity due to geometrical effects. Using this simple formula the increase of emissivity by simply drilling a cavity inside a solid body can be calculated. In Fig. 20 this calculation was graphically shown for some values of wall emissivities and ratios of length to radius of a cavity. Eq. (2.1) takes only into account the ratio of the radius to the length of the cavity, neglecting the form of the bottom or the temperature distribution or other additional effects, which might influence the cavity emissivity. More sophisticated methods have been described in the literature. In the next section the method for calculating the cavity emissivity used in the present work is described in detail.

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Fig. 21. Schematic representation of a cavity radiator for generation of blackbody radiation.

2.1. Blackbody emissivity According to Kirchhoff’s considerations the thermal radiation generated by isothermal walls forming an enclosure is blackbody radiation. To get access to this radiation a small hole in the envelope must be opened as shown in Fig. 21. As indicated in Fig. 21 a photon which enters this cavity will undergo several reflections at the cavity walls and the probability for the escape of this particular photon is relatively low. Therefore, the cavity has a high absorption. The only radiation emitted by this cavity is the radiation generated at the cavity walls. As long as the temperature at the walls is the same everywhere inside the cavity the radiation emitted by this hole will be a near ideal approximation of blackbody radiation. The approximation is improving the smaller the area of the opening is in comparison to the inner surface area of the enclosure. To quantify the accuracy of the blackbody radiation the concept of emissivity has been introduced. The emissivity is a measure of the deviation of the ability of a real surface to emit thermal radiation in comparison to an ideal blackbody, i.e. an emissivity of unity describes an ideal blackbody. The emissivity of a real surface is in principle a function of wavelength, temperature, and angle of view. For measuring the emissivity the ratio of the spectral radiance of the source with respect to the radiation of an ideal blackbody at the same temperature has to be measured. However, if the emissivity is close to unity a measurement can not easily be performed with the lowest uncertainties. In those cases the calculation of the emissivity by theoretical considerations based on the simple model introduced above is necessary. A variety of mathematical models has been developed either based on Monte Carlo simulations or on the multiple reflection method [38]. The analytical models commonly contain a series of contributions, which have to be summed up and thus make most of them impractical. Recently, commercial raytracing software applying the Monte Carlo method became available. Such software makes it possible to investigate the emissivity of isothermal blackbodies with a complicated geometry [39]. For practical blackbodies, especially for application in remote sensing instrument calibration, a large aperture area is required, which leads to a non-isothermal temperature distribution on the cavity walls due to reasonable radiation loss through the large aperture. This effect significantly influences the emissivity of the blackbody and has to be considered in high-accuracy metrological measurements. Bedford and co-workers [40] addressed this topic long ago, adapting the series method for use with non-isothermal blackbodies. A recent approach in calculating the emissivity of a non-isothermal blackbody with the Monte Carlo method was presented by Prokhorov [41]. However, there is still need for comparisons of calculated effective emissivities of non-isothermal blackbodies with complicated cavity geometries and corresponding experimental data. The method described here is based on a Monte Carlo simulation, which allows the calculation of the emissivity of cavities with non-isothermal temperature distributions on the cavity walls. First the calculation of the emissivity for the isothermal case is described, which is then extended for non-isothermal blackbodies. In this approach the inverse ray tracing method is applied and photons are considered which enter the cavity through the aperture instead of photons emitted by the blackbody [42]. A photon is sent into the cavity at a certain angle and its trace through the cavity is followed by geometrical calculations. Only emission angles permitted by the experimental situation are considered. Once the photon reaches the cavity wall, a random number generator is used to determine the subsequent path of the photon. For the simulation diffuse reflection according to Lambert’s cosine law for the inner surface of the blackbody is assumed. With a probability of 1 − εw (λ) the photon is reflected from the surface in a randomly distributed direction, with εw (λ) being the local emissivity of the cavity wall. The photon is traced until it is either absorbed or escapes the cavity. The isothermal emissivity εiso is simply the ratio of absorbed photons and photons sent into the cavity. This emissivity, εiso , depends only on the geometry of the cavity and the local emissivity εw (λ) of the cavity walls. For the emissivity εnoniso (λ), which depends on the non-isothermal temperature distribution of the cavity walls, every absorbed photon is weighted with the spectral radiance according to Planck’s law with the temperature T of the location where the photon is absorbed. The spectral radiance Lλ (λ, T ) assigned to a single absorbed photon is connected to the spectral radiance Ls,λ (λ, T ) of a blackbody at the same temperature by Lλ (λ, T ) = εw (λ)Ls,λ (λ, T ).

(2.2)

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For the slightly different temperatures T − ∆T in a non-isothermal cavity, Eq. (2.2) can be rewritten in linear approximation as

 ∂ Ls,λ (λ, T ) · ∆T . Lλ (λ, T − ∆T ) = εw (λ)Ls,λ (λ, T − ∆T ) = εw (λ) Ls,λ (λ, T ) − ∂T 

(2.3)

Using Wien’s approximation for Planck’s law, the derivative is

∂ Ls,λ (λ, T ) c2 = Ls,λ (λ, T ), ∂T λT 2

(2.4)

so that Eq. (2.3) becomes approximately c2

h

i ∆ T Ls,λ (λ, T ). 2

(2.5)

Lλ (λ, T − ∆T ) = εw (λ)ε∆T ,w (λ, ∆T ) Ls,λ (λ, T ) ,

(2.6)

Lλ (λ, T − ∆T ) = εw (λ) 1 −

λT

Eq. (2.5) can be written as with ε∆T ,w (λ, ∆T ) specifying the change in local emissivity due to a temperature difference ∆T . Eq. (2.6) describes the spectral radiance contributing from the special trace of one photon. In order to calculate the spectral radiance for the whole cavity, the contributions of all photons have to be summed up and the sum has to be divided by the number of contributing photons N. This procedure can be applied for both isothermal (Eq. (2.7)) and non-isothermal (Eq. (2.8)) cavities Lcav,iso =

1 X N

Lcav,noniso =

F (Ωn ) εw (λ)Ls,λ (λ, T ) ,

(2.7)

n

1 X N

F (Ωn ) εw (λ)ε∆T ,w (λ, ∆Tn ) Ls,λ (λ, T ) .

(2.8)

n

In Eqs. (2.7) and (2.8), F (Ωn ) denotes the view factor for the trace of photon n. The product F (Ωn )εw (λ) is implicitly taken into account by the Monte Carlo simulation described above. The emissivity ε∆T describes the transition from an isothermal cavity to a non-isothermal cavity defined by

ε∆T =

Lcav,noniso Lcav,iso

.

(2.9)

On the other hand, with the previously calculated emissivity εiso for an isothermal cavity, the spectral radiance of the isothermal cavity can be written as Lcav,iso = εiso Ls,λ (λ, T ) .

(2.10)

Therefore, the emissivity εnoniso of a non-isothermal blackbody is

εnoniso = εiso · ε∆T = =

1 X N

n

Lcav,noniso Ls,λ (λ, T )



=

F (Ωn ) εw (λ) 1 −

1 X N c2

λT

F (Ωn ) εw (λ)ε∆T ,w (λ, ∆Tn )

n

 ∆ T . n 2

(2.11)

In radiation thermometry the reference temperature T is usually the temperature of the bottom of the cavity. This is the quantity that is measured by a radiation thermometer. Using the approach described here, the emissivity of cavity radiators with arbitrary temperature distributions along the cylindrical walls can be calculated. 2.2. Consideration of the effective refractive index of blackbody radiation In optical measurements the index of refraction of the medium the optical radiation is passing through has to be considered. Especially in the case where the optical radiation passes through different media this fact is of high concern, as also refraction at the contact surface between the two media has to be considered. Working in laboratory conditions, the medium the radiation is passing through is ambient air and the formulas for the index of refraction of air derived by Edlén and modified by others can be used to calculate the refractive index with low uncertainty [43–47]. However, the situation is more complex when dealing with the measurement of blackbody radiation generated in a cavity radiator. Due to the different temperature inside and outside the blackbody cavity and due to purging of the cavity with argon or other inert gases, the index of refraction of the media inside and outside the blackbody cavity are, in general, different. In literature usually simply the index of refraction of ambient air is taken into account in the optical path between the opening of the cavity and the detecting instrument. For standard reference laboratory conditions the refractive index of ambient air is shown in Fig. 22 [44]. However, this accounts only for the travelling part of the optical radiation outside the cavity, neglecting

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221

Fig. 22. Refractive index of air at 1013 hPa, 0.03% CO2 content, 50% relative humidity and 23 ◦ C [126] (courtesy Optical Society of America).

all effects originating inside the cavity. In addition some radiometric and photometric experiments reported in literature the refractive index is often neither stated nor even considered. For the generation of the blackbody radiation inside the cavity the index of refraction of the medium inside the cavity has to be taken into account [32] and this might differ significantly from the value of the ambient medium, especially if the blackbody cavity is at elevated temperatures or purged by an inert gas. Planck’s formula for the spectral radiance Ls,ν of a blackbody emitted per unit area and frequency interval in a solid angle is given as a function of frequency ν as [32] Ls,ν =

2hn2 c2 exp khνT

ν3
. −1

(2.12)

In Eq. (2.12), n is the index of refraction of the medium inside the cavity at the cavity temperature T , ν is the frequency of the optical radiation, h is Planck’s constant, k is the Boltzmann constant and c the velocity of light in vacuum. For optical radiation inside a medium with refractive index n the following equation for the wavelength λ, the velocity of light in the medium cn and the frequency ν holds cn =

c n

= νλ,

(2.13)

resulting in c . (2.14) nλ Rewriting Eq. (2.12) as a function of wavelength λ inside the medium requires a relation for Ls,λ and Ls,ν . This can be found by using Eq. (2.14) resulting in the following expression

ν=

Ls,ν

dLs dλ

dλ = = = Ls,λ , dν dλ dν dν dLs

i.e. Ls,λ

dν = Ls,ν dλ

dν = c . with dλ nλ 2

(2.15)

Using Eqs. (2.15) and (2.14), Eq. (2.12) can be written as follows

Ls,λ

  dν 2hc c 2hc 2 3 λ nλ2 dλ n n2 λ 5       = = . = exp k nhcλ − 1 exp k nhcλ − 1 exp k nhcλ − 1 T T T 2hn2 c 3 c 2 n3 λ3

(2.16)

Eq. (2.16) describes the spectral radiance at the exit aperture of a cavity radiator containing a medium with refractive index n inside as a function of the wavelength λ inside this medium. Eq. (2.16) is identical to the results given in literature [32]. The radiation emitted by the opening of the blackbody cavity has to travel the distance between the exit aperture and the detector through the ambient medium, usually air at ambient conditions. This will cause a wavelength shift according to the index of refraction of the ambient medium, namb . To account for this, Eq. (2.16) is rewritten in terms of vacuum wavelength λ0 = n · λ resulting in 2hc 2 λ5 n2 5o

2hn3 c 2

n

Ls,λ0 =

 exp

hc λ k n n0 T

λ50

=

 −1

exp



hc

k λ0 T



. −1

(2.17)

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Accounting for the wavelength shift in the ambient medium requires replacement of λ0 by namb · λ 2hc 2

2hn3 c 2 n5amb λ5o

Ls,λ = exp



hc k namb λT



n3 n2amb amb n3

= −1

exp



hc knamb λT

λ5



.

(2.18)

−1

Comparing Eq. (2.18) with Eq. (2.16) reveals that the effect of including the transmission of the radiation through the ambient medium with refractive index namb introduced the factor (n/namb )3 , changing the effective spectral radiance Ls,λ by this factor. Another effect which has to be accounted for is the reflection occurring at the contact area between the media inside of the blackbody cavity and the outside media, usually ambient air. For perpendicular incidence the corresponding reflectivity R is

 R=

namb − n namb + n

2

.

(2.19)

In Eq. (2.19) it is assumed that the transition between the two media inside and outside the blackbody is abrupt. However, in reality, as it is a transition between two gaseous phases, the transition is not abrupt but smooth and diffuse. Therefore, Eq. (2.19) is an upper limit for the expected reflectivity. As the two media inside and outside the blackbody cavity are usually both gases the reflectivity at the contact between these two media will be negligible small. Therefore, Eq. (2.19) as a worst case assumption will be used. Consideration of Eq. (2.19) leads to the final formula for the spectral radiance emitted by a blackbody radiator at temperature T with a medium of refractive index n into a medium of refractive index namb

" Ls,λ =

1−



namb − n

2 #

n3amb exp

namb + n

with the factor a (namb , n) =

2hc 2 n2amb λ5

n3

 1−





namb −n namb +n

hc knamb λT

2 



2hc 2 n2amb λ5

= a (namb , n) −1

n3 , n3amb

exp



hc knamb λT



,

(2.20)

−1

which accounts for the effect of the two different media inside

and outside the blackbody cavity. For the correct consideration the factor a(namb , n) has to be taken into account, when calculating the spectral radiance emitted by a blackbody radiator according Planck’s law. For an estimation of the error introduced by simply using Eq. (2.16) with the refractive index of the ambient medium instead of using Eq. (2.20) calculations have been performed. The most pronounced effect is expected, if the refractive index inside and outside the blackbody cavity differs significantly. This will be the case for blackbodies at elevated temperatures. Usually such blackbodies are operated in an argon gas atmosphere [48]. As the refractive index will approach unity for highest gas temperatures, for a worst case estimation it is assumed that the refractive index of the argon gas inside a blackbody cavity at temperatures of 3273 K is unity. Using the modified Edlén equation for the refractive index of air as the outside medium given in Ref. [44] the relative difference between Eq. (2.20), considering the two different media inside and outside the blackbody cavity correctly, and Eq. (2.16), considering only the refractive index of ambient air, has been calculated. The resulting relative difference as a function of wavelength is shown in Fig. 23. Fig. 23 shows that neglecting the difference in the refractive index inside and outside the cavity will introduce a nearly constant relative difference of about 0.08% over the whole wavelength range from 200 nm to 15 µm, which is theoretically increasing to 0.1% at wavelengths down to 100 nm, well within the vacuum ultraviolet regime. This error neglecting the different refractive indices might be tolerable in some photometric experiments. However, it has to be taken into account for accurate radiance temperature measurements. The introduced error in determining the temperature of a blackbody at 3273 K using radiometric methods is also shown in Fig. 23. In this worst case estimation this error is in the order of several tenths of a kelvin in the visible, increasing to several kelvin in the infrared region. In literature it is stated that the obtained uncertainty in radiometric temperature measurements nowadays is as low as a few tenths of a kelvin for temperatures around 3273 K [49–51]. The correct consideration of the refractive index is, therefore, mandatory for radiometric temperature measurements. This is especially important for the determination of the thermodynamic phase transition temperatures of the novel metal-carbon and metalcarbide-carbon eutectics for an improved International Temperature Scale, where an uncertainty in temperature of about 0.1 K is required [52]. 2.3. The blackbody radiators Several types of blackbodies have been developed for application in radiation thermometry and are reviewed here for the sake of completeness. This overview is restricted to the temperature range usually covered by commercial radiation thermometers and important for radiometry and photometry, i.e. from 200 K up to 3273 K. For the temperature range from 200 K up 1235 K the blackbodies are described in brief and only the blackbodies for temperatures above 1300 K are described in more detail.

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Fig. 23. Relative difference (solid line) between the spectral radiance of a blackbody radiator with the correct consideration of the medium with refractive index of unity inside and of ambient air outside (according Eq. (2.20)) with respect to the spectral radiance only considering the refractive index of the ambient air (according Eq. (2.16)). Also shown is the resulting error (dashed line) in determining the radiance temperature of a blackbody at 3273 K (a) overview, (b) detail at short wavelengths; the arrow in the graphics indicates that the right ordinate is valid for the dashed line while the left ordinate is valid for the solid line [126] (courtesy Optical Society of America).

2.3.1. Low temperature blackbodies The blackbodies best suited for the temperature range from 200 K up to 1235 K are heat-pipe blackbodies. The cavity of this type of blackbodies is formed by a heat-pipe. A heat-pipe is a doubled walled tube filled with a small amount of a substance. The pressure inside the tube is adjusted so that the liquid and the gaseous phase of the substance are in equilibrium [53]. The effect of the heat-pipe is the following. At a hot spot in the cavity an additional amount of the liquid substance evaporates lowering the temperature due to the loss of evaporation heat. At cold spots part of the gaseous phase condenses and heats this location by delivering condensation heat to this point. Due to this effect the temperature homogeneity of the cavities formed by heat-pipes is extremely good. However, the heat-pipe effect only works in a restricted temperature range. Therefore, four different heat-pipe materials for the temperature rang 200 K up to 1235 K are applied. In the temperature range from 200 K up to 320 K Ammoniac is used as a heat-pipe material. A design of this type of blackbody is shown in Fig. 24. The cavity of this blackbody is covered by a special high emissivity coating having an emissivity of about 0.95 [54]. The large cavity together with the black velvet coating results in an emissivity of 0.9999. For temperatures below 273 K the cavity has to be purged by dry nitrogen to prevent ice forming inside the cavity. For the adjacent temperature range from 320 K to 540 K the heat-pipe material used is water. The design is similar to the ammonia heat-pipe blackbody shown in Fig. 24, but with a smaller dimensions of 420 mm in length. Due to the higher temperatures a different high emissivity coating is used with a smaller emissivity of about 0.9. The smaller length and the lower emissivity coating leads to an effective emissivity of 0.9998. For the temperature above 540 K another blackbody design shown in Fig. 25 is applied. Due to the high temperatures no high emissivity coating can be used. The intrinsic emissivity of the cavity material, r sandblasted and oxidised Inconel , is approximately 0.75. Together with the geometrical dimensions shown in Fig. 25 an emissivity of 0.9996 is obtained. In this temperature range two heat-pipe materials are applied, these are caesium (540 K up to 920 K) and sodium (770 K up to 1234 K). The details of these blackbodies are summarized in Table 2.1. Traceability of the radiation temperature to the ITS-90 for the four above mentioned blackbodies is given by standard platinum resistance thermometers (SPRT), which are inserted into the heat-pipe close to the bottom of the cavity. Optimized controlling of the temperature in connection with the high thermal mass of the blackbodies yields a temperature stability to

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Fig. 24. Cross section of the ammonia heat-pipe blackbody used with the low temperature calibration facility of the PTB [51].

Fig. 25. Cross section of the heat-pipe blackbodies for temperature above 540 K up to 1235 K [51]. Table 2.1 Details of the used blackbodies Black body

Temperature range (◦ C)

Heat-pipe material Length of cavity (mm)

Diameter of cavity (mm)

Wall emissivity

Blackbody emissivity

NH3 H2 O Cs Na

−60 to +50

Steel Titanium Inconel 600 Inconel 600

60 60 41 41

0.965 0.88 0.75 0.75

0.9999 ± 0.00006 0.9998 ± 0.00015 0.9996 ± 0.00017 0.9996 ± 0.00017

50 to 270 270 to 650 500 to 962

525 420 368 368

These blackbodies are applied at the low and medium temperature facility of the PTB and are schematically shown in Fig. 26.

within ±10 mK. These blackbodies are used as standards for temperature radiation for calibration of radiation thermometers and thermal imagers. Using a high quality transfer radiation thermometer, they can also serve to calibrate customer’s blackbodies. When radiation thermometers or thermal imagers are calibrated they are fixed on a xyz translation stage, which allows computer controlled positioning over 3000 mm, 600 mm, and 200 mm, respectively, with at least 50 µm accuracy and 10 µm reproducibility (see Fig. 26). 2.3.2. High-temperature blackbody For higher temperature application above the silver point, special high-temperature blackbodies of two different types are utilized. These blackbodies are now described in more detail. 2.3.3. High-temperature furnace with carbon cavity Both types apply a cavity formed by graphite. The first one uses a simple graphite tube to form the blackbody cavity as shown in Fig. 27. As can be seen from Fig. 27 the cavity is radiative heated by a surrounding, meander like heating element. The heating element is manufactured from semi-rigid C–C composite and has slots cut into it to increase the electrical resistance. This simple geometry enables a quasi homogeneous temperature distribution over a length of about 100 mm. However, in this furnace the cavity can only be as large as 27 mm in diameter, restricting the available space for handling high-temperature

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225

Fig. 26. A schematic experimental arrangement of the low and medium temperature calibration facility of the PTB [51].

Fig. 27. Picture of the interior of the high-temperature furnace manufactured by Nagano. The meander like heating element around the supporting furnace cavity can clearly be seen.

fixed-point cells. This furnace can reach temperatures as high as 3100 K. The whole furnace can be evacuated and purged by argon or helium. When measurements are required the window in front of the cavity is removed to enable direct assess to nearly ideal blackbody radiation. This furnace is manufactured by Nagano Incorporation Japan and is hereafter designated the Nagano furnace. The Nagano furnace is mainly used for high-temperature fixed-point measurements with incorporated fixed-point cells. Therefore, a detailed radiometric characterisation of its radiating behaviour is not necessary. Some details of such a characterisation can be found in Ref. [55].

2.3.4. High-temperature furnace with pyrolytic graphite cavity The second type of furnace is quite different in design. The blackbody cavity is formed by pyrolytic graphite rings as shown in Fig. 28. This special type of graphite has a layered structure with a very high thermal and electrical conductivity inside the layer and a very small thermal and electrical conductivity in the perpendicular direction (Fig. 29).

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Fig. 28. Inner cavity of the HTBB formed by pyrolytic graphite rings.

Fig. 29. Schematic of the pyrolytic graphite structure and the directions of high and low thermal and electrical conductivity.

The advantage of using pyrolytic graphite is twofold. At first, due to the layered structure a good electrical heating and a good homogeneous temperature distribution is obtained. Second, the pyrolytic graphite is extremely inert with respect to oxidisation, enabling considerable operation times at highest temperatures. The design of the cavity is shown in the schematic cross section of the high-temperature blackbody given in Fig. 30. The pyrolytic graphite rings are clamped together by a spring from the backside and directly heated by an electrical current. The diameter of the radiating cavity is about 37 mm and its length is about 200 mm. Applying an electrical current of about 650 A temperatures up to 3200 K can be reached. Due to the pyrolytic structure the graphite rings are highly inert against oxidisation. This allows the blackbody to be used the whole time without a window by simply purging the cavity with pure argon gas. The application without a window provides unhindered access to the thermal radiation from the cavity without any deviation introduced by absorption due to a window with limited transmission. Due to the large inner diameter of the cavity, this type of blackbody can be used as radiometry and photometry standard also without the use of high-temperature fixed-point. For such an application a detailed radiometric characterisation is necessary to obtain the accurate emissivity of the blackbody cavity [128]. One main effect influencing the emissivity is the temperature distribution across the cavity walls and has, therefore, be measured. For determining the cavity wall temperature of the blackbody cavity the used radiation thermometer was tilted

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227

Fig. 30. Cross section of the pyrolytic high-temperature blackbody [48,130].

Fig. 31. Geometry of the temperature measurement of the cavity walls.

Fig. 32. Obtained temperature distribution across the cavity walls of the HTBB [128].

by 5◦ with respect to the symmetry axis of the cavity and moved with respect to the blackbody to keep the cavity wall in the focus of the pyrometer for every measurement point according Fig. 31. Due to the limiting opening aperture of 26 mm in diameter only a section of 120 mm in length of the cavity wall beginning from the bottom could be investigated. The cavity temperature was sensed in steps of 5 mm. Fig. 32 shows the resulting radiance temperatures of the cavity wall for several cavity temperatures from 1337 K up to 3200 K. All measured temperature distributions show a distinct temperature rise with respect to the temperature of the bottom of the cavity, followed by a temperature drop towards the cavity opening. The qualitative explanation for this behaviour is that the bottom is cooled by radiation losses through the aperture as is also the case for the area near the opening. The width of this temperature change decreases with increasing temperature, while its height only slightly increases with increasing temperature. The temperature was sensed at two opposite locations at the cavity walls, and no significant difference was found between the two results, indicating that the measured temperature variation has a rotational symmetry. The wall temperatures measured by the radiation thermometer were effective radiance temperatures, which were determined by the temperature of the wall in the field of view of the pyrometer and the reflected radiative power

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Fig. 33. Measured and calculated radiance temperature distribution along the cavity wall at a temperature of 2300 K [128].

contributing from other parts of the cavity. For extracting the true wall temperature at the location of measure a ray-tracing Monte Carlo simulation program based on the theory described above was used [42]. The photons were inserted into the cavity according to the actual geometric configuration of the experiment and their path was determined assuming diffuse reflection at the cavity walls with a probability of 1 − εw (εw = 0.9 emissivity of the graphite material of the cavity wall) until they were ‘‘absorbed’’. Photons leaving the cavity without being absorbed were not considered. The emissivity of the iso-thermal cavity is then simply the ratio of the absorbed and the inserted number of photons, resulting in an emissivity of 0.999. In case of a non-isothermal cavity temperature distribution the contributing photons were additionally weighted by the spectral radiance of the corresponding surface element according Planck’s law of radiation for the temperature at the location of absorption. The temperature of the cavity bottom serves as the reference temperature. In general, an iterative approach has to be followed, starting with an initial temperature distribution, followed by a numerical calculation of the obtained radiance temperature distribution, comparing the results with the experimental data and start again with a slightly improved temperature distribution. Fortunately, the measured radiance temperature is close to the true temperature of the cavity wall, as long as the emissivity of the cavity wall is sufficiently high. In case of the HTBB the cavity wall emissivity is assumed to be 0.9, which ensures a rapid convergence. In order to obtain an analytical expression of the cavity temperature variation, the experimentally determined radiance temperatures were interpolated using a polynomial of third order. A typical result obtained for a cavity temperature of 2300 K is shown in Fig. 33. The coefficients of the third order polynomial were used to extrapolate the temperature distribution across the whole cavity length, resulting in a typical temperature decrease of several hundreds kelvin at the cavity opening. These calculated temperature distributions were used to determine the overall cavity emissivity of the blackbody, the so-called effective emissivity, at different temperatures, considering the geometry of the measurement with the radiation thermometer. As Planck’s law of radiation is strongly dependent on wavelength this is also the case for the effective emissivity. The obtained effective emissivities are presented as a function of wavelength in Fig. 35. Taking the temperature of the bottom of the cavity as a reference temperature, the temperature increase near the bottom of the cavity results in effective emissivities higher than one, rapidly increasing for decreasing wavelength. Although the temperature decreases dramatically towards the aperture of the cavity, which should lower the effective emissivity, this effect is compensated by the temperature increase near the cavity bottom. This is the case, because due to the optical elements used in the radiation thermometer only photons generated at the bottom of the cavity were measured by the radiation thermometer. This can be seen in Fig. 34 where the path of the photon measured by the radiation thermometer is graphically presented [56]. It can be seen from Fig. 34 that most of the reflections of the photon happens in the rear part of the cavity. Even though the maximum temperature increase near the bottom is much smaller than the temperature drop near the aperture, it is dominating the effective emissivity, because for the chosen geometry using the radiation thermometer most of the detected photons will be emitted in the rear part of the cavity. The results presented in Fig. 35 clearly indicate that the non-isothermal distribution of the temperature of the cavity walls critically affects the effective emissivity and, with it, the apparent temperature of the blackbody. Typically blackbodies cavities are used as reference radiation sources and their temperature is either determined by contact thermometry or by radiometric methods. In both cases the determined temperature generally applies for an isothermal cavity at a given wavelength. In case of a non-isothermal cavity, the temperature determined in such a way is valid for a specific wavelength only. For example, in the case of the HTBB determining the blackbody temperature of 1337 K by absolute filter radiometry, at an effective wavelength of 676 nm, results in an error of about 0.8% in spectral radiance when observing the blackbody at a wavelength of 200 nm. Figs. 36 and 37 show the resulting error in temperature, when the cavity temperature has been radiometrically obtained at 676 nm or at 1000 nm and has been applied at another wavelength. The resulting temperature error may be as large as 0.6 K, clearly indicating that a correction with respect to the non-isothermal temperature of the cavity walls is essential for use of the high-temperature blackbody BB3200pg as high-accurate radiation source in the visible and UV spectral range. The results show that it is necessary to carefully investigate the temperature uniformity of blackbodies to obtain the best possible results.

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Fig. 34. Graphical representation for the path of a photon inside a cavity calculated for the geometry of the radiation thermometer used [56].

Fig. 35. Calculated effective emissivities including the non-isothermal behaviour of the wall temperature for different blackbody temperatures as a function of wavelength. The temperature of the cavity bottom was taken as reference temperature [128].

Fig. 36. Resulting temperature correction using the cavity temperature obtained at a wavelength λo = 676 nm at different wavelengths λ [128].

However, such investigations can only be performed, if the temporal temperature stability of the investigated blackbody and the size-of-source effect of the applied radiation thermometer are sufficiently good. Special care should be taken when the temperature of a blackbody is measured with a multi-wavelength radiation thermometer, which only works properly, if the emissivity of the blackbody does not vary with wavelength, which is only the case for an isothermal blackbody

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Fig. 37. Resulting temperature correction using the cavity temperature obtained at a wavelength of λ0 = 1000 nm at different wavelengths λ [128].

Fig. 38. Schematic drawing of a precision aperture.

cavity. Taking into account the non-isothermal temperature distribution of the HTBB cavity the radiance temperature of this blackbody can be determined with lowest uncertainties. 3. Radiometric precision apertures For absolute measurements of optical radiation the geometry of the experiment must be known very accurately. To define the solid angle and the field of view precision apertures are used. The principle features of such apertures are shown in Fig. 38. The inset of Fig. 38 shows the detailed structure of the edges, which are the most important features to ensure that the apertures can be used in precise radiometry. An ideal knife edge structure, as thin as possible together with a 90◦ angle of the remaining cylindrical part – the land – ensure an effective optical area independent of the angular distribution of the incoming radiation as shown in Fig. 39. For imperfect edges as shown on the left in Fig. 39 the collected radiation after the aperture is strongly dependent on the distance between aperture and detector. In contrast for ideal edges as shown on the right in Fig. 39 the collected radiation is independent on the distance between the aperture and the detector. These effects are more pronounced in case of divergent or convergent radiation which is frequently encountered in radiometry and photometry. To obtain a nearly perfect edge structure the land of the aperture must be as thin as possible, in the order of 10 µm–20 µm and as steep as possible. Such an edge is not easily obtainable. One of the promising technical solutions is diamond turning of the apertures. An electron microscope was used to investigate the edge structure in detail. Some results obtained on different apertures are shown in Fig. 40. The results of the diamond turning production of precision optical apertures critically depend on the material used and the experience of the manufacturer. It is quite difficult to obtain structures like those shown in Fig. 40(c) and (d). The ideal material for those apertures is aluminium or copper. Once such nearly ideal apertures have been obtained the effective radiometric area has to be determined before use in absolute radiometry and photometry.

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Fig. 39. Dependence of the effective optical area on the edge structure.

Fig. 40. Diamond turned apertures (a), (b) non-ideal edges, (c), (d) nearly ideal edges.

3.1. Absolute determination of the effective optical area of the apertures Due to the fragile edges the area of the precision radiometric apertures can not be measured by contact methods, noncontact optical methods have to be used. The method used here applies a sensor based on a laser diode, emitting radiation at a wavelength of 780 nm. The laser beam is focused on the sample surface, and the reflected radiation is collected by a photodiode. A movable lens keeps the focus of the laser radiation at the sample surface. The position of the lens provides a measure of the topography. The system works properly if at least 12% of the total emitted laser radiant power is reflected by the surface. A signal indicates that the 12% reflection threshold is reached. In a former approach this signal was used for indicating the position of the aperture edge [57]. The aperture is fixed on a rotation stage, which is mounted on a xy stage with sub-micrometer resolution. An interferometer with spatial resolution of 0.1 µm measures the position. Rotating the aperture enables measurements of diameters at different angular positions. The position of the edges is determined by moving the aperture with respect to the sensor. Starting from the middle of the aperture the position of the edges is obtained by the reflection threshold signal at the two opposite end points of a diameter. With respect to the central maximum of the Gaussian radiant power distribution of the laser beam, the position of the reflection threshold deviates by half the reflection threshold diameter of the laser focus as shown in Fig. 41.

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Fig. 41. Definition of threshold diameter and explanation for the deviation of actual laser position and location of the detected edge.

Fig. 42. Method for measuring the diameter of the precision optical apertures [61] (courtesy IOP Publishing Ltd.).

A correction needs to be applied for the diameter of the laser beam spot. A chromium-on-glass structure with known dimensions was used for that purpose, yielding the diameter of the laser focus at the reflection threshold for chromium. Aluminium, the material of the apertures under investigation, has different optical properties, which has to be taken into account by an additional correction. This procedure has two main disadvantages. First, the laser focus calibration has to be repeated for every aperture built of a new material with different reflection properties. Second, the uncertainty resulting from the unknown laser focus diameter proved to be the main contribution to the overall uncertainty of the first approach. Nevertheless, the technique was sufficiently accurate for absolute radiometry experiments in the early 1990’s [57]. To overcome the problems of different surface reflectivities an improved approach was developed which uses the full measured reflection signal, instead of the single value detected by the built-in reflection threshold criterion. The situation of the measurement can be seen in Fig. 42. As the edge of the aperture approaches the Gaussian laser profile, the reflected radiant power increases, until the whole Gaussian laser profile is reflected. For an infinitely sharp edge, its position and the maximum of the Gaussian laser profile is reached at half the value of the full reflection signal. This criterion is currently used to detect the position of the aperture edge with enhanced accuracy. For an analytical description we assume that the Gaussian laser beam profile is centred on the z-axis. The aperture covers the left side of the xy plane, with the edge parallel to the y-axis at position x. Then, the reflected radiant power S (x) is S (x) =

x

Z



S0

π

−∞

1/2 a

exp −

ξ2 a2



dξ =

S0 2

+

S0

x erf . 2 a

(3.1)

S0 is the full reflected radiant power, a is the 1/e diameter of the Gaussian laser beam, and erf denotes the error function [58]. The error function can be approximately evaluated using its series expansion, S (x) =

S0 2

" 1+

2

∞ X

π 1/2

n=0

# (−1)n  x 2n+1 . n! (2n + 1) a

(3.2)

In Eq. (3.2), x is replaced by x − x0 , because the experimental aperture position is not measured from the centre of the laser focus, but from some other fixed point. Then, Eq. (3.2) is used to perform a least squares fit to the experimental data and to obtain simultaneously values for the laser focus diameter, the reflection amplitude and the edge position.

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Table 3.1 Applied corrections and uncertainty budget for the aperture diameter measurement (values for apertures with nominal diameter d = 5 mm) Source of uncertainty Air temperature Atmospheric pressure Humidity Abbe error, vertical Abbe error, horizontal Angle error Cosine error Centring error Laser wavelength 1 dim approximation Partial fitting error Random uncertainty Fitting error Sum (50%, theor.) Bevel uncertainty Sum (50%, exp.)

Correction of d in nm

−50 1 0.3 0.1 50

−250

−249 −249

u in nm

Type

1 1 2 10 10 1 0.2 0.1 0.1 50 30 80 50 112 280 302

B B B B B B B B B B B A A B

The position of a perfect edge coincides with the position where the reflected radiant power reaches half the maximum value, so that the reflection threshold criterion should be replaced by a 50% reflection criterion. Using the 50% reflection criterion eliminates the need of corrections of the measured aperture diameter with respect to the unknown laser focus diameter. Application of this technique to apertures made of highly reflecting materials may result in a reflected laser radiant power saturating the detector system. If this happens before the maximum reflected radiant power is reached, the position of the edge cannot be determined by the 50% reflection criterion, because this would yield unrealistically small diameters (Fig. 42). In this situation only the rising part of the measured reflected radiant power signal before saturation is fitted. The contributions to the uncertainty of the method are given as standard uncertainties (k = 1). Since the same experimental set-up as for the former approach was used, the uncertainties arising from the set-up are the same as described in Ref. [57] and are discussed only briefly for an aperture of 5 mm nominal diameter. Distance measurements using optical interferometers require determination of air temperature, pressure and humidity. Uncertainties of 0.3 K for the air temperature, 0.1 kPa for the pressure and 30% for the relative humidity were taken into account. As the aperture edge positions are measured at opposite sides of a diameter, misalignment of the aperture surface with respect to the translation stage causes an ‘‘angle error’’ by measuring only a projection of the diameter. A similar ‘‘cosine error‘‘is introduced if the translation stage does not move parallel to the laser beam of the interferometer. If the laser beam of the interferometer does not coincide with the trace of the laser focus of the microtopography sensor on the aperture surface and the motion of the translation stage is not rectilinear, the so-called ‘‘Abbe offset error’’ occurs [59]. This error was minimised by decreasing the offset between the interferometer beam and the trace of the focus for the horizontal part. For the vertical part, this was not possible, so that a corresponding correction was applied, which has a relatively large uncertainty. In the first approach, the main uncertainty contribution was connected with the separate determination of the radius of the Gaussian laser profile, introducing an uncertainty contribution of 0.3 µm for the aperture diameter. In our new approach, the separate determination of the laser beam radius is not required and this uncertainty is absent. However, there are new uncertainty contributions instead, arising from the fitting procedure of the measured reflection signals. One contribution arises from neglecting the curvature of the aperture edge and the two-dimensional extension of the laser focus. Numerical investigations of the corresponding error showed that for the laser focus diameter used, the relative deviation of the determined aperture diameter is about 1 part in 105 . A corresponding correction of 0.05 µm was applied, and this correction was taken as uncertainty. All uncertainties mentioned thus far are of type B. Another contribution is brought about by the fitting procedure, particularly if the detector system saturates before the maximum reflected radiant power is reached. Using the chromium-on-glass structure, which shows no saturation, we compared partial fitting results with the results obtained by fitting the whole reflection signal. This investigation showed that a major correction of −0.25 µm with an uncertainty of 0.03 µm has to be applied to the aperture diameter if the diameter is derived from partial fitting results. There are two type A (i.e. statistically derived) uncertainty components: (a) the uncertainty of the fit of the measured reflection signals and (b) the random uncertainty of the scatter in the measured diameters. A summary of all the corrections and uncertainties for the new procedure is given in Table 3.1. Compared with the best combined standard uncertainty of 0.3 µm obtained at ideal apertures for the former approach, the new 50% reflection criterion results in three-fold enhanced accuracy. As an additional advantage, the correction is only −0.25 µm for the 50% reflection criterion instead of 1.15 µm for the formerly used reflection threshold criterion. The resulting relative combined standard uncertainty of the aperture diameter is 2 × 10−5 for apertures with 5 mm diameter and the resulting relative combined standard uncertainty of the area is 4 × 10−5 .

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Fig. 43. Measured and fitted reflected radiant power for the chromium-on-glass structure together with their derivatives [61] (courtesy IOP Publishing Ltd.).

These uncertainties would be reached for apertures with perfect edges. The measured aluminium apertures exhibit bevels and other structures at the very edges. Edges of this kind are not well defined, and a ‘‘bevel uncertainty’’ has to be taken into account. Investigation of the edges of our aluminium apertures by electron microscopy revealed a bevel uncertainty of 0.28 µm. Due to this additional uncertainty contribution, the reduction of the total uncertainty for our aluminium apertures is less pronounced than the reduction expected for apertures with more perfect edges. Therefore, the main advantages of the new procedure for the measured apertures are the independence of the aperture material and the small overall correction. For testing the new evaluation procedure we measured the chromium-on-glass structure, which consists of a sequence of chromium stripes separated by stripes of bare glass. A recorded reflectivity curve obtained at a glass-to-chromium edge and its derivative is presented in Fig. 43. Also shown are a fit to the experimental data according to Eq. (3.2) and its first derivative, representing the laser radiant power in the direction of measurement. The agreement of experimental and fitted curves is a proof for the accuracy of our approach. For the laser radiant power profile we obtain a 1/e radius of (0.676 ± 0.001) µm from the derivative of the fit and of (0.65 ± 0.17) µm from the derivative of the signal. The measured dimensions of the chromium-on-glass structure have a relative deviation of less than 10−4 from the results obtained by an interferometer comparator [57]. The diameters of several apertures were determined by the former reflection threshold method and the new 50% reflection method. These results were compared with measurements performed with the reflection threshold method over 10 years ago. The corresponding results and the respective uncertainties are shown in Fig. 44 for two representative apertures. It is evident from Fig. 44 that the aperture diameters obtained between 1991 and 2006 are well within the uncertainty of 0.3 µm, i.e. differ by less than 6 × 10−5 for the 5 mm aperture and less than 2 × 10−5 for the 20 mm aperture. 3.2. Relative determination of the area of the precise apertures The absolute measurement method requires a large number of measurements of the diameter of the apertures to determine the average diameter and therefore the aperture area. In general, a number of 150–600 diameter measurements have to be performed. This is usually done overnight and one aperture can be measured in one night. To facilitate the monitoring of the aperture areas a relative method has been developed, which significantly reduces the measurement time to less than one hour. The experimental set-up is schematically depicted in Fig. 45. The main parts are an optical bench of five meter length, an irradiance source of high temporal stability, a detector, which can either be a trap detector or an integrating sphere detector. The stable and homogeneous irradiation field is produced using a FEL lamp of 1 kW radiant power. The whole optical bench, including the lamp and the detector shown in Fig. 45 are enclosed in a light-tight housing. The stability of the FEL lamp is critical for the measurement. This was checked by measuring the photocurrent of a trap detector monitoring the FEL lamp over more than one hour. The result of this check is shown in Fig. 46. It can be seen that the stability of the FEL lamp is better than 10−3 over that period of time. The optical path of about 4 m length is equipped with several baffles and screens to suppress stray-light. The rather large optical path has been chosen to produce an illumination at the apertures matching as close as possible the situation for their later use, ensuring that the radiometric area, which may differ from the geometrical area [60], is measured. This procedure is especially necessary for ultra-small apertures as it can not be assumed that the edges of such apertures are ideally steep. The trap detector is provided with a high precision aperture of 5 mm nominal diameter, which has been absolutely calibrated using the method described in Section 3.1. The edges of the reference aperture were carefully investigated and have been found to be nearly ideally

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Fig. 44. Results of the aperture diameters measured from 1991 until 2006 for a nominal 5 mml (a) and a 20 mm (b) diameter aperture.

Fig. 45. Scheme of the experimental set-up of the aperture comparator [127] (courtesy IOP Publishing Ltd.).

Fig. 46. Relative change of the photocurrent measured by a trap detector monitoring the FEL light source as a function of time [127] (courtesy IOP Publishing Ltd.).

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steep [57,61]. Therefore, the area of the reference aperture is nearly independent on the divergence of the illumination, i.e. the geometrical area and the radiometric area are identical. The apertures under investigation were positioned in front of the trap detector using a translation stage. The whole detector system, including the apertures under investigation, is embedded in a light tight enclosure and separated from the rest of the optical set-up by a baffle. Application of trap detectors for comparative apertures area measurements has already been reported in literature, however, these measurements have been restricted to apertures with areas of several millimetres in diameter, where the trap detector was used only as an alternative detector [60]. Here we describe the application of trap detectors to the measurement of ultra small area apertures, which makes extensively use of the out-standing properties of the trap detector. The advantage of using a trap detector instead of an integrating sphere detector system is twofold. First, due to the multiple reflections of the incoming radiant power and the large inner surface of the integrating sphere, the fraction of the optical radiation reaching the detector diode is very low resulting in a poor signal to noise ratio. A trap detector is built of three photodiodes, arranged in such a way that the incoming radiation undergoes five reflections before it re-emits the trap [see Section 4.3 and Ref. [62]]. Due to the five reflections the total reflectivity of the trap detector is very low (<0.01 in the visible). Therefore, virtually the whole radiant power is collected resulting in relatively high signals, even for ultra small aperture areas. Second, due to the configuration of a trap detector the spatial homogeneity of this sensor is of the order of 10−4 in the visible spectrum, as has been verified by laser-based measurements [62,63]. It is therefore as good as or even better than the homogeneity of an integrating sphere. When ultra small aperture areas are measured, spatial homogeneity is essential to obtain reliable results. The low total reflectivity of the trap detector ensures that only a negligible part of the light is re-emitted and no inter-reflection between the detector and the back side of the apertures is expected. However, the restricted angle of acceptance of the trap detector requires an irradiation with low divergence (smaller than 2◦ in our case). But this can easily be obtained by using a long distance between radiation source and aperture surface. For the actual measurement the photocurrent of the trap detector covered with the reference aperture and the photocurrent of the trap detector with the aperture under investigation in front of the reference aperture is measured using a current to voltage converter [64] and a high precision digital voltmeter. Due to the differences in aperture areas when measuring ultra small apertures, large ratios of photocurrents have to be measured. This was accommodated by using several high-quality resistors, ranging from 1 k to 10 G with steps in the power of ten for transforming the current to voltage. The absolute ratio of two adjacent resistors was determined with the same measurement system by determining the signal of one aperture with two different resistors. For the actual measurement, the resistors have been chosen so that the same sensitivity range of the voltmeter can be used. Due to the experimental arrangement in the case of measuring ultra small apertures, the aperture under investigation and the reference aperture can not be positioned at the same distance from the irradiance source, but were about 5 mm apart. For a distance of lamp to trap aperture of about 3.5 m, this accounts to a relative correction of 1.0014 of the signal ratio. The calculation of the aperture area was performed by averaging at least five single signal ratio measurements. The influence of the remaining stray-light was considered for every measurement by blocking the direct light path and performing the same measurement procedure once again. Finally, at least three measurements at different locations of the aperture under investigation with respect to the centre of the reference aperture were averaged to get the final value for the aperture area and to rule out slight in-homogeneity effects. Measuring precision apertures with lowest uncertainties, especially with ultra-small areas, requires consideration of diffraction effects, which will limit the range of application. After passage through an ultra-small aperture, the optical radiation is diffracted and forms a so-called Airy pattern, which consists of several maxima and minima [65]. A typical measure of the radius of the Airy pattern is the radius a of the central maximum, which can be calculated as [65] a=

3.83 Rλ 2π

r

,

(3.3)

with r being the radius of the aperture, R the distance behind the aperture, and λ the wavelength of the radiation. The distance R between the aperture and the location of the last reflection inside the trap detector is 90 mm. Due to the tilting, the active area of the photodiode has a radius of 6.5 mm. Using an average wavelength of 500 nm for calculation purposes yields about 10 µm for the smallest measurable aperture diameter. However, only 84% of the total radiant power is included in the central maximum of the Airy pattern [65]. For collecting more than 99.5% of the radiant power more than 41 rings of the Airy pattern have to be integrated. However, even for the first diode, which is only 30 mm behind the ultrasmall apertures, this only is the case for apertures with a diameter larger than 140 µm. Therefore, a correction to the signals measured for ultra-small apertures has to be added. This correction has been calculated by numerical integration of the Airy pattern and consideration of the five reflections inside the trap detector. The resulting fraction of the total radiant power, which is not collected by the trap detector for the ultra-small apertures in the range from 40 µm to 300 µm follows the function

∆s s

= 1.3038 (2r )−1.0761 ,

(3.4)

with r being the radius of the aperture given in µm. Eq. (3.4) gives the fraction of radiation passing the apertures, which is not collected by the trap detector yielding a smaller effective aperture diameter of the aperture under calibration.

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Table 3.2 Diameters d of the apertures measured with the optical microscope and the focused laser technique Aperture (µm)

d (microscope) (µm)

d (laser) (µm)

d (comparator) (µm)

300 270 170 50

276.6 ± 2.5 249.8 ± 3.1 164.6 ± 3.0 –

– 250.0 ± 0.2 163.6 ± 4.4 43.5 ± 0.5

279.0 ± 0.6 251.3 ± 0.6 164.5 ± 0.4 44.3 ± 0.2

Two to four diameters in perpendicular directions have been averaged to get the final diameter. The values given in the last column were calculated from the area of the apertures obtained by the new comparator technique, assuming the aperture areas to be ideally circular. The uncertainties given for the microscope and the focused laser technique result from the statistical scatter of the measurements, while for the comparator technique the uncertainties according Table 3.3 are given for a coverage factor of k = 1.

According Eq. (3.4) only apertures larger than 4 mm yield a deviation smaller than 5 × 10−5 , which can be neglected. However, the signals obtained for smaller apertures have to be corrected for using Eq. (3.4), and the uncertainty of the correction is estimated to be 20%. The relative correction obtained according Eq. (3.4) are 2.2%, 0.5%, and 0.3% for apertures with diameters of 50 µm, 170 µm, and 300 µm, resulting in a correction of about 1 µm for all the mentioned aperture diameters. In principle, application of Eq. (3.4) allows for the measurement of apertures with diameters well below 50 µm, however, a practical limit is expected for diameters below 20 µm, as for smaller apertures the uncertainty introduced by the correction exceeds the uncertainty due to the experimental set-up. As an example, ultra small apertures used for the normal incident set-up at the Berlin synchrotron radiation source BESSY II for calibration of deuterium lamps in the wavelength range from 40 nm to 400 nm [66] were investigated. They were lithographically processed in a thin (approx. 50 µm thick) nickel foil. The apertures investigated are of circular shape and have nominal diameters of 300 µm, 280 µm, 170 µm, and 50 µm. Due to their small diameter application of the above mentioned non-contact techniques results in high uncertainties. Nevertheless, two of these techniques were applied to the apertures, namely the optical microscope and the focused laser technique described in Section 3.1. Both measurements were performed to get diameters of the aperture in two to four directions only, and these values were averaged to obtain the final diameter of the circular aperture area. The apertures have subsequently been measured with the new comparator technique using a high-precision reference aperture, which has been calibrated with the absolute technique [61]. The diameter of this aperture has been determined to (5.0292 ± 0.0003) mm. Although the comparator technique is a direct measure of the area, in the following only diameters, calculated by assuming the apertures to be ideally circular, were presented to allow an easy comparison with the results obtained by the other techniques. The results of the three different measurement techniques are shown in Table 3.2 The uncertainties for the new comparator technique given in Table 3.2 are explained in detail, while the uncertainties given for the two other methods mainly result from the average of the diameter measurements in only two to four directions. The three techniques agree within their combined uncertainties. Note that the results obtained with the comparator technique are average diameters, because it can not be assumed that the apertures are ideally circular. The equation for determination of the area of the unknown aperture applying the comparator technique is Aunknown = Aref Iunknown /Iref

(3.5)

with Aunknown and Aref being the area of the unknown and the reference aperture, and Iunknown and Iref being the photocurrent measured either for the unknown or the reference aperture. The contributing uncertainties arise from the uncertainty of the area of the reference aperture and the uncertainty of the photocurrent measurement. The latter one is split in the uncertainty of the detecting trap detector, the short time stability of the lamp and the uncertainty of the resistors. Table 3.3 summarizes the uncertainty contributions, given at the coverage factor k = 1 and the overall uncertainty of the set-up. One key point in determining the ultra-small aperture areas using the comparator method is the measurement of large ratios of photocurrents. Therefore, a highly linear detector system is essential. The applied detector system has been in use for a long time for the absolute calibration of the spectral responsivity of trap detectors with respect to the primary cryogenic radiometer [67], and has carefully been checked for its linearity [68,69]. The obtained uncertainty has been included in Table 3.3 as uncertainty of the resistor ratio. For the smallest aperture of about 50 µm another major contribution to the uncertainty is due to the diffraction correction, which has been described in detail above. The time delay between the measurements of reference aperture and unknown aperture is rather long due to a slow motion translation stage, introducing a considerable large uncertainty for the lamp stability. Additionally, the exact position of the apertures with respect to each other can only be determined with an absolute uncertainty of about 1 mm in the present set-up, resulting in a similar high uncertainty contribution. Up to now, no temperature control of the detector box is used, resulting in a temperature change during the measurement as high as 1 K, what is also a source of high uncertainty. All of the last three contributions can easily be reduced at least by a factor of 2, which should lead to an overall uncertainty of about 10−3 .

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Table 3.3 Uncertainty budget of the comparator method at a coverage factor of k = 1 Contribution

50 µm Relative correction

Reference aperture Reference area Temperature stability (∆T = 1 K ) Stability of the lamp Photocurrent Diffraction correction Uncertainty of diffraction correction Spectral responsivity of trap Homogeneity of trap Temperature coefficient of trap (∆T = 1 K) Resistor ratio of the I/U-converter Geometry Repeatability Sum

2.2 × 10−2

170 µm Relative uncertainty 0.08 × 10−3 0.08 × 10−3 0.02 × 10−3 1.0 × 10−3 4.5 × 10−3

Relative correction

250 µm and 300 µm Relative uncertainty

Relative uncertainty

4.4 × 10−3 0.5 × 10−3 0.2 × 10−3 1.0 × 10−3

1.0 × 10−3 0.5 × 10−3 0.2 × 10−3 1.0 × 10−3

0.08 × 10−3 0.08 × 10−3 0.02 × 10−3 1.0 × 10−3 1.3 × 10−3 – 0.6 × 10−3 0.5 × 10−3 0.2 × 10−3 1.0 × 10−3

0.05 × 10−3 1.0 × 10−3 1.0 × 10−3 4.8 × 10−3

0.05 × 10−3 1.0 × 10−3 1.0 × 10−3 2.3 × 10−3

0.05 × 10−3 1.0 × 10−3 1.0 × 10−3 2.2 × 10−3

5 × 10−3

0.08 × 10−3 0.08 × 10−3 0.02 × 10−3 1.0 × 10−3 1.5 × 10−3

Relative correction

3 × 10−3

4. Spectral responsivity of detectors used in this study The geometry of an optical radiation measurement is depicted in Fig. 18 and consists of three major parts: the radiation source, the radiation detector, and the geometric details of the set-up. Measuring optical radiation can be performed in several ways, depending on different physical properties. One key property needed is the spectral responsivity of the radiation detector. Among the applications of silicon trap detectors is the determination of thermodynamic temperature by means of absolute radiometry. This requires accurate spectral responsivity calibration. Existing spectral responsivity scales [70–72] were not able to meet the requirements of advanced thermodynamic temperature measurement. Therefore, at PTB the spectral responsivity scale, based on the laser-operated radiation thermometry cryogenic radiometer (RTCR) [68] has been developed. In addition the scope has been extended to the ultra violet part of the spectrum. Laser-based calibrations of the spectral responsivity in the UV are only available at a few laser lines and their uncertainty is about an order of magnitude larger than the uncertainty of calibrations in the visible. Nevertheless, this is still sufficient to establish an UV responsivity scale with reduced uncertainty [73]. When trying to extend and improve the spectral responsivity scale, one has to consider three main areas: (i) the optical power measurement with the cryogenic radiometer, (ii) the characterisation of the transfer detectors, and (iii) the interpolation of the spectral responsivity between the laser lines. Techniques developed in these three areas are now described. 4.1. Optical power measurement with the radiation thermometry cryogenic radiometer (RTCR) The RTCR and the automated calibration procedure are described in detail in Ref. [68]. The lasers employed are a Kr+ laser, an Ar+ laser with optional intra-cavity frequency-doubling, and a continuously tuneable Ti:sapphire ring laser. The lasers provide 12 lines in the UV (238 nm–400 nm), 14 lines in the visible (400–799 nm) and continuously tuneable radiation in the NIR (800 nm–1015 nm). At present, 5 wavelengths in the UV and 15 wavelengths in the visible and NIR are used for calibrations (Fig. 47). The wavelength of the tuneable Ti:sapphire ring laser is measured by a wavemeter with an uncertainty of 1 part in 106 and is stable within 0.01 nm during an overnight calibration. The maximum radiant power used for calibration is 50 µW below 300 nm, 400 µW between 300 nm and 450 nm, and 500 µW above 450 nm. The laser radiation is power-stabilised by an electro-optical modulator and expanded to a beam diameter of 4 to 6 mm using a telescope and a spatial filter. Two quadrant photodiodes are used to quantify the scattered light hitting the detectors to be calibrated, but missing the cavity of the RTCR. Since the quadrant photodiodes are not sensitive at wavelengths below 350 nm, a spherical mirror with a central hole of the same diameter as the cavity entrance aperture is used outside the cryostat to focus the scattered light onto a windowless UV-sensitive Si photodiode. This new way of measuring the scattered light, which is described in more detail for the NIR wavelength region in Ref. [74], is significantly more reliable than the previous extrapolation method. The uncertainty of the optical power measurement with the RTCR is 0.02%–0.08% in the UV and 0.007%–0.01% in the visible and NIR. 4.2. Characterisation of transfer detectors Because of the higher absorption coefficient and shorter penetration depth, the electrical and optical properties of the Si surface and the passivating layer are more important and have a greater influence on the properties of photodiodes in

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Fig. 47. Measured spectral responsivity of two Si trap detectors calibrated at the RTCR at laser wavelengths in the visible and NIR and in the UV [63] (courtesy IOP Publishing Ltd.).

the UV than in the visible. As a result, Si photodiodes cannot be regarded as stable in the UV. Exposure to radiation with a wavelength below 300 nm changes the responsivity not only at the wavelength of exposure, but also at shorter and longer wavelengths even in the visible [75]. Detailed studies of this UV instability have shown that Hamamatsu S1337-1010 and S5227-1010 photodiodes are stable within 0.3% for wavelengths above 238 nm and a radiant exposure below 10 J/cm2 , but are not suited for shorter wavelengths [75,76]. Therefore, two sets of three-element reflection trap detectors are used as transfer detectors at the RTCR. One set of trap detectors built of windowless Hamamatsu S1337-1010 and S5227-1010 photodiodes are calibrated in the UV and sometimes in the visible and NIR. The other set with large S3411 photodiodes are only used in the visible and NIR and thus not effected by UV exposure. In the NIR, especially at photon energies comparable to the energy of the band gap, the penetration depth of the radiation can exceed the thickness of the diode. In this case, the electrical and optical properties of the backside strongly influence the properties of the photodiode. Therefore, the non-linearity, uniformity, and temperature dependence of the responsivity of photodiodes and trap detectors have been investigated especially in the UV and NIR. 4.2.1. Non-linearity The non-linearity of a detector with respect to the incoming optical flux is a measure for the deviation from a linear dependence of generated photocurrent from the incoming optical flux. The non-linearity of the detectors has been studied using the classical flux addition technique described in detail in Ref. [77]. A laser beam is split into two beams, A and B, of about the same power with orthogonal polarisations. Blocking one or the other of the beams or superimposing them on the detector, the photocurrents IA , IB , and IAB are measured. The non-linearity N is N (IAB ) = (IAB − IA − IB )/IAB .

(4.1)

Investigations of Hamamatsu S1337 photodiodes from 250 nm to 800 nm showed that this type is linear within 1 × 10−4 for photocurrents below 0.3 mA [77]. Recent measurements confirmed this result for trap detectors with large S3411 photodiodes at 647 nm, but showed strong supralinearity at 900 nm and especially at 1000 nm (Fig. 48), which correspond to photons with energies comparable or slightly lower than the energy difference between electrons in the conduction band and valence band of silicon. In all non-linearity measurements the beam diameter was 3.2 mm (1/e2 ) which is about the beam diameter used during calibration at the RTCR. The (absolute) uncertainty uN of the results for N shown in Fig. 48 is 2×10−5 . If the trap detector was calibrated at a photocurrent Ical the following correction has to be applied to the responsivity at a photocurrent Iuse = Ical 2−n : s(Iuse ) = s(Iuse × 2n )/

n Y 

1 + N (Iuse × 2k ) .



(4.2)

k=1

At λ = 1000 nm, with √ Ical = 0.2 mA, Iuse = 0.2 nA and n = 20, one obtains s(Iuse ) = s(Iuse 2n )/(1.0021 ± 0.0001) with an uncertainty of uc = uN n for this correction. 4.2.2. Spatial uniformity The spatial uniformity of a detector is a measure for the spatial dependence of its spectral responsivity across its sensitive area. The uniformity of the trap detectors was investigated by scanning with a focused laser beam at 1000 nm, 514 nm and 257 nm. At 1000 nm and 514 nm, a diameter of 0.3 mm and a step width of 0.25 mm were chosen. At 257 nm, the diameter

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Fig. 48. Non-linearity of a Si trap detector at two wavelengths in the NIR [63] (courtesy IOP Publishing Ltd.).

and the step width were both increased to 0.5 mm to avoid UV induced responsivity changes. The peak-to-peak variation of the spectral responsivity of the detector in the central area of 6 mm diameter is 0.1% at 1000 nm, 0.02% at 514 nm and 0.2% at 257 nm for new or seldom used trap detectors. Trap detectors with S3411 photodiodes delivered in 1992 showed a larger non-uniformity at 1000 nm. In contrast to diodes delivered in 1995, these older diodes have a rather non-uniform backside without additional finishing. As a considerable fraction of the 1000 nm radiation reaches the backside, the non-uniformity of the backside influences the uniformity of the diode responsivity. The non-uniformity can be larger by an order of magnitude or more for detectors which have been intensively used over several years. The data of the scans were used to estimate the contribution of the non-uniformity to the total uncertainty in the trap calibration. 4.2.3. Temperature coefficient The temperature coefficient (TC) is a measure of the dependence of the spectral responsivity on the temperature of the detector. The temperature coefficient was determined by recording the response of the trap detector to power-stabilised laser radiation while changing the detector temperature using a temperature-controlled water jacket. The temperature was measured with a standard uncertainty of 0.05 K with a Pt100 sensor located on the aluminium mount of the photodiodes. Fig. 49 shows the TC of a trap detector at several wavelengths between 467 nm and 1015 nm and the TC of a single S1337 photodiode between 257 nm and 800 nm [68]. The single photodiode TC is larger than that of a trap detector because the main part of the TC of a single photodiode arises from reflectance changes [78] to which trap detectors are less sensitive. Around 1000 nm the trap responsivity starts to decrease and the TC increases strongly with wavelength. The TC reaches 0.45%/K at 1015 nm, about 500 times the value around 500 nm. The reason for this is that the band-gap energy of silicon decreases with increasing temperature and the wavelength-dependent responsivity curve in Fig. 47 is shifted towards shorter wavelengths. The wavelength dependence of the TC can be reproduced by a simulation using the model of quantum efficiency described later and taking the temperature dependence of the band gap [63] into account (broken lines in Fig. 49). The high TC around 1000 nm requires accurate control and measurement of the detector temperature during calibration and application. If the uncertainty contribution of the temperature correction is to be kept below 0.006% at 1000 nm, the temperature has to be stable within 0.02 K. This is difficult but achievable with the temperature control arrangement at PTB. 4.2.4. Long term stability Long term stability is a measure for the stability of the spectral responsivity of a detector over a time frame of a year. Because trap detectors have been calibrated at the RTCR since 1992, their long-term stability can be evaluated and systematic changes can be determined even it is lower than the measurement repeatability on a one-year time scale. Fig. 50 shows the relative changes in responsivity of several traps at different wavelengths, indicating that the changes are stronger at shorter wavelengths. 4.2.5. Spectral responsivity uncertainty at distinct laser lines The individual uncertainties affecting a responsivity calibration of a Si trap detector at laser lines in the spectral range from 238 nm to 1015 nm are shown together with the combined total uncertainty in Fig. 51. The contribution arising from the instability of trap detectors at short wavelengths is given for a period of one month, which is about the time necessary to

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241

Fig. 49. Temperature coefficient of a Si trap detector and a single Si photodiode (taken from [4]) in the 250 nm to 1015 nm spectral range. The temperature coefficient calculated with the model of Section 4.3 is shown as a dotted line [63] (courtesy IOP Publishing Ltd.).

Fig. 50. Long-term changes of the spectral responsivity of trap detectors calibrated at the RTCR at three wavelengths [63] (courtesy IOP Publishing Ltd.).

carry out the calibrations in the visible and NIR. Depending on the application, additional contributions for the non-linearity and the stability may also have to be taken into account. The total relative standard uncertainty of the spectral responsivity is u = 0.01% (k = 1) at the laser lines in the visible and near IR below 950 nm. At longer wavelengths the uncertainty increases up to 0.026% due to the increasing temperature coefficient of the Si trap detectors. The low uncertainty of u = 0.1% (k = 1) in the UV made it possible to establish a continuous UV spectral responsivity scale with low uncertainty [73].

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Fig. 51. Contributions to the total uncertainty of the spectral responsivity at laser lines and of the spectral responsivity scale [63] (courtesy IOP Publishing Ltd.).

4.2.6. Model of quantum efficiency and spectral responsivity scale In order to obtain a reasonable function for interpolation of the spectral responsivity between the laser wavelengths we used a model for the quantum efficiency of a silicon trap detector originally developed by Gentile et al. [72] and extended to longer wavelengths for our purpose. The internal quantum efficiency ηi (λ) of a silicon photodiode of thickness h can be calculated from [79]

ηi (λ) =

h

Z

exp(−α(λ)x)α(λ)P (x)dx,

(4.3)

0

if the absorption coefficient α(λ) and the collection efficiency P (x) are known. The collection efficiency describes the fraction of radiation capable of generating electron–hole pairs in the silicon photodiode. It is a function of the distance from the diode surface. A simple model shown in Fig. 52 starts with a collection efficiency Pf at the SiO2 –Si interface. The value of Pf is mainly influenced by trapped surface charges and lattice deformation due to the mismatch of the lattice constants of SiO2 and Si. Further inwards, the collection efficiency increases linearly with depth and reaches its maximum of unity at the location of the pn junction of the diode at depth t. Then, a linear decrease follows until Pb , the bulk value of silicon, is reached at depth D. This value is maintained up to the rear surface of the diode at depth h. For wavelengths longer than 950 nm, the Si absorption coefficient α(λ) is very small and the penetration depth of photons becomes larger than the thickness of the silicon diode. The photons are partly reflected at the rear surface of the diode and penetrate the diode once again. With an additional term taking account of this effect, the total internal quantum efficiency ηi of the diode becomes

ηi (λ) = Pf +

1 − Pf

(1 − exp[−α(λ)t ]) −

1 − Pb

α(λ)t α(λ) [D − t] − Pb exp[−α(λ)h] + r exp[−α(λ)h]Pb .

(exp[−α(λ)t ] − exp[−α(λ)D]) (4.4)

Here, r is the reflection coefficient of the back side of the photodiode. r is assumed to be independent of wavelength, because for typical thicknesses of the investigated photodiodes only photons with a wavelength larger than 900 nm will reach the backside of the photodiode. The resulting spectral responsivity s(λ) is s(λ) =

(1 − r (λ)) ηi (λ)nair λe hc

,

(4.5)

with nair the index of refraction of air, e the electron charge, h Planck’s constant, c the velocity of light and λ the wavelength in air of the radiation. The reflectivity r (λ) of the trap detector is approximated by the analytical function r (λ) = a exp(b/λ) + c λ + d.

(4.6)

The quantities a, b, c and d were used as additional fitting parameters. Evaluation of Eq. (4.4) for the internal quantum efficiency requires knowledge of the absorption coefficient of silicon. Optical constants for silicon from 0.6 nm up to 333.3 µm are found in Ref. [80]. Unfortunately, little data is available for the wavelength range of interest, in particular between 900 and 1100 nm. Moreover, the values given in Ref. [80] differ from the values used by Geist et al. [81] in the long wavelength range. However, as we are not interested in absolute values for the internal quantum efficiency, but in an accurate spectral responsivity scale, the knowledge of the absolute value of the

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243

Fig. 52. Collection efficiency of a silicon pn photodiode as a function of distance from the front surface [125] (courtesy Optical Society of America).

absorption coefficient is not needed, only its evolution as a function of wavelength must be known to obtain a reliable fit. For this purpose, a sufficiently precise interpolating function for the absorption coefficient α(λ) for silicon for wavelengths larger than 400 nm was found to be

α(λ) = A1 exp(A2 /(λ − λ0 )) + A3 λ + A4 λ−1 + A5 ,

(4.7)

with the best fit coefficients A1 = 0.53086 µm−1 ,

A2 = 0.469643 µm, λ0 = 0.256897 µm, A3 = −0.28801 µm−2 , A4 = −0.988739, A5 = 0.282027757 µm−1 . Our model accurately represents measured spectral responsivity data over the wavelength range from 400 to 1015 nm. In Fig. 47, the solid line represents a fit according Eq. (4.5). There is excellent agreement of the fit to the experimental data. While the deviation of the fit from the measured data is smaller than 0.003% for the wavelength interval 400–950 nm, it increases to 0.04% when fitting the whole wavelength range up to 1015 nm. This is due to the lack of and the uncertainty in the absorption data for wavelengths longer than 1000 nm. For testing the quality of the interpolation we removed single calibration points and refitted the remaining points. Using this interpolation for calculation of the spectral responsivity at the omitted wavelength, the result was well within the uncertainty of the calibration in the wavelength range 476 and 900 nm, i.e. better than 0.01%. At longer wavelengths this kind of test was not possible, as the material properties of silicon change very rapidly with wavelength in this region and every calibration point is thus necessary to determine the fit parameters. In conclusion, the only additional contribution to the uncertainty of the interpolation considered is the deviation of the interpolation from the measured data at the calibration points. The total standard uncertainty of the responsivity scale including the uncertainty of the interpolation between the laser lines is shown in Fig. 51. Between 400 and 950 nm, the uncertainty of the scale is only slightly increased by the interpolation and is 0.011%, but due to the rapidly varying material properties the interpolation significantly increases the overall uncertainty of the scale to 0.05% for wavelengths larger than 984 nm. Through these investigation the spectral responsivity scale based on the radiation thermometry cryogenic radiometer (RTCR) operated with laser radiation has been further improved and extended. Five laser wavelengths in the UV (238–400 nm) and another fifteen in the visible and near IR (400–1015 nm) cover almost the entire spectral range where silicon photodiodes can be used in air. The total relative standard uncertainty of the spectral responsivity at the laser lines is u = 0.01% (k = 1) in the visible and near IR below 950 nm. The uncertainty increases up to 0.026% at 1015 nm because of the large temperature coefficient of the trap detectors. In the UV a relative standard uncertainty of 0.1% or lower is achieved at the laser lines. With an improved model for interpolation between the laser lines, a continuous scale of spectral responsivity has been established in the wavelength range from 400 to 1015 nm. 4.3. Model for the temperature dependence of the spectral responsivity In recent years the demand for high accuracy spectral responsivity scales increased steadily, especially in the field of radiation temperature measurements. For example, the fabrication of semi-conducting devices applies rapid thermal annealing processes, which require temperature control within 1 K even at temperatures higher than 1000 ◦ C [82]. For the purpose of radiation temperature measurements, silicon-photodiode-based systems are widely used in science and industry [83]. The reason for this is the possibility of predicting the spectral responsivity of silicon by analytical models and their good temporal stability [83–85,63].

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Fig. 53. Schematic view of a reflection Trap detector; also shown is the path of the light inside the Trap [125] (courtesy Optical Society of America).

The spectral responsivity s(λ) of photodetectors depends on several experimental parameters and can be described by Eq. (4.5). The main parameter of the spectral responsivity is ηi (λ), the internal quantum efficiency of e.g. silicon photodiode detectors, i.e. the ratio of the number of photoelectrons contributing to the measurable signal obtained from the photodiode and the number of photons reaching the detector. The internal quantum efficiency of silicon photodiodes can be described analytically as a function of wavelength in the range 400–1020 nm by Eq. (4.6). To reduce the influence of the reflectivity R and to further increase spatial homogeneity and temporal stability of the detector, special radiation trapping arrangements of silicon photodiodes were developed, the so-called trap detectors [86]. The simplest of these consist of three silicon photodiodes arranged so that the incoming radiation is reflected from the first diode to the second and to the third one and then the same way back (see Fig. 53). Thus, the radiation undergoes five reflections before leaving the detector. This yields a reflection coefficient for this kind of trap detector of the product of the five single reflection coefficients, also significantly reducing the effect of variations in reflectivity. As the photodiodes in a trap detector are connected in parallel, and their shunt resistances are sufficiently high, the trap detector behaves like a single photodiode in all other respects, but has a reflection coefficient of the fifth power of that of a single diode. Due to their linear spectral response silicon photodiodes are ideal detectors for the wavelength range from 400 to 950 nm [83,86]. Recently, increasing efforts have been expended in extending radiometric scales into the near- and mid-infrared region to meet the requirements of calibration for remote sensing instruments [87]. While the spectral characteristics of silicon photodiodes are well known in the wavelength range from 400 to 950 nm, this was not the case for photon energies near the band gap, i.e. in the wavelength range around 1000 nm. In this spectral region the absorption coefficient of silicon is rather low, resulting in a large penetration depth of the photons. In addition, the temperature dependence of the absorption coefficient of silicon increases significantly with increasing wavelength, critically affecting the spectral responsivity [63]. For further improvement in the accuracy and reliability of spectral responsivity scales in the visible and near-infrared, it is essential to accurately determine the temperature dependence of the spectral responsivity of the photodiodes and to understand the underlying physical mechanisms theoretically. Consequently, the aim of this investigation was twofold. First, a model of the temperature dependence of the spectral responsivity was developed, and the influence of the influence parameters is investigated. Second, theoretical predictions based on this model are compared with experimental results obtained for silicon reflection trap detectors. According to Eq. (4.5) the observed temperature dependence of the spectral responsivity s(λ) may originate from the temperature dependence of (i) the reflectivity R of the detector (TCR ), (ii) the index of refraction of air nair (TCN ), and (iii) the internal quantum efficiency ηi (λ) (TCQ ). Hence, the relative temperature dependence for s(λ) is ds sdT

=

−dR dηi dnair + + . ηi dT nair dT (1 − R) dT

(4.8)

In the following the three contributions and the quantum efficiency are treated separately. The TCR of silicon photodiodes has been measured at room temperature for 476 nm (at an incident angle of 0◦ ), 633 nm, and 1047 nm (at an incident angle of 65◦ ) [88,89]. The obtained relative changes in reflectivity of a single silicon photodiode with temperature are tabulated in Table 4.1. As described above, the reflectivity of a trap detector is the product of the reflection coefficients for the five reflections inside the trap detector. Assuming that the temperature dependence of these five reflections is the same for all angles of incidence, the relative temperature dependence of the reflectivity is five times larger for a trap than for a single diode. However, the trap reflectivity is smaller than 0.01 for a wavelength of 400 nm and smaller than 0.0025 for wavelengths larger than 800 nm, as compared to the reflectivity of about 0.3 for a single diode [86]. The trap reflectivity enters the spectral responsivity via (1 − RTrap ), and the temperature coefficient of (1 − RTrap ) is shown in Table 4.2.

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Table 4.1 Relative change of reflectivity of a single silicon photodiode with temperature Wavelength (nm)

dR/dT /R (K−1 )

476 633 1047

1.43 × 10−4 5.3 × 10−5 1.78 × 10−4

Table 4.2 Trap reflectivity and temperature coefficient of (1 − RTrap ) calculated using RTrap = R5 and the values of Table 4.1 for R dRTrap

dRTrap

(K−1 )

Wavelength (nm)

RTrap

(1−RTrap )dT

476 633 1047

0.00454 0.00271 0.00198

3.26 × 10−6 7.20 × 10−7 1.77 × 10−6

(1−RTrap )dT

dR / RdT

0.023 0.014 0.010

Also given is the ratio of the trap temperature coefficient and that of a single photodiode.

The TCN can be calculated using the expression for nair (T ) given by Birch and Downs [90]. The corresponding temperature coefficient in Eq. (4.8) is dnair nair dT

P −10−8 Pa · 0.00972

= −8

1 + 10



0.601 − 0.00972

(T +273.15) ◦C



− P Pa

0.0036616 1 + 0.0036610

(T +273.15)

.

(4.9)

◦C

Here P denotes air pressure measured in Pa and T is the air temperature measured in kelvin. The TCQ has to be calculated. Using the expression of ηi (λ) given in Ref. [63] (see also Eq. (4.4)) 1 − Pf 1 − Pb (1 − exp[−α(T )t ]) − (exp[−α(T )t ] − exp[−α(T )d]) α(T ) · t α(T )(d − t ) + Pb d(r − 1)e exp[−α(T )h],

ηi (λ) = Pf +

(4.10)

and assuming that α(T ) is the only temperature-dependent parameter in this expression, the derivative of ηi (λ) with respect to temperature can be written as dηi

 1 − Pf 1 − Pf 1 − Pb = − exp[−α(T )t ] + (1 − exp[−α(T )t ]) + (exp[−α(T )t ] − exp[−α(T )d]) 2 dT α(T ) · t α(T ) α(T )2 (d − t )  1 − Pb dα − . (4.11) (−t exp[−α(T )t ] + d exp [−α(T )d]) − hPb d(r − 1)e exp[−α(T )h] α(T )(d − t ) dT

In Eqs. (4.10) and (4.11) Pf and Pb are the collection efficiencies of silicon at the SiO2 /Si interface and in the bulk, respectively. The dimensions t, d, and h are shown in Fig. 52 and denote the depth of the pn-junction, the end of the depletion region and the thickness of the photodiode, respectively [72]. r is the reflection coefficient for radiation at the back side of the diode, which has to be included for photon wavelengths longer than about 900 nm [63]. According to Eq. (4.11), the temperature dependence of the internal quantum efficiency is determined by the temperature dependence of the absorption coefficient (TCA ). In the following, three different models for the latter quantity are applied, depending on the wavelength range to be considered. Model I The TCA for semiconductors is correlated to the temperature dependence of the band gap energy Egap (T ), which can be described analytically [91]. The band gap of silicon varies with temperature according Egap (T ) = Egap (0 K) −

aT 2

( T + b)

,

(4.12)

with Egap (0 K) = 1.17 eV, a = 4.73 × 10−4 eV/K and b = 636 K. For photon energies hν in the vicinity of the band gap, α(T ) can be approximated by [91]


γ α(T ) ∼ hν − Egap (T ) .

(4.13)

For silicon, an indirect band gap semiconductor, the exponent is γ = 2 [91]. This results in a relative temperature dependence of α(T ) of



2

2 (T2aT − (TaT +b ) +b)2

1 dα  =  α dT hν − Egap (0 K) −



aT 2 (T +b)

 .

(4.14)

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Fig. 54. Temperature coefficient for the absorption coefficient of silicon according the different models. Circles: model I [91], squares: model II [92], line: model III [93]. [125] (courtesy Optical Society of America).

Eq. (4.14) can be expected to provide a good approximation for the temperature dependence of α(T ) for wavelengths around 1000 nm. Model II For the temperature dependence of α(T ) at shorter wavelengths, results by Jellison and Modine [92] are available. They found that α(T ) of silicon for wavelengths below 750 nm can be modelled by

α(T ) = α0 (λ) exp(T /T0 (λ)),

(4.15)

α0 (λ) and T0 (λ) are tabulated in Ref. [92]. This model yields a constant temperature coefficient for a fixed wavelength: 1 dα 1 = . α dT T0

(4.16)

Model III The models leading to Eqs. (4.14) and (4.16) are only valid in restricted wavelength ranges. In order to obtain a more general valid model covering the whole wavelength range of interest, we use the empirical formula for the TCA given by Weakliem and Redfield [93]. This formula is obtained by shifting a reference curve α(E , T1 ) = α(hν, T1 ) = α(hc /λ, T1 ) to

α(E 0 , T2 ) = α(E + ∆E , T1 )

(4.17)

with E 0 = E + ∆E

∆E = c (E )(T2 − T1 ) and c (E ) = 1.2 × 10−3 eV/K, c (E ) =



1.3

E eV

for E > 1.7 eV



− 1.0 × 10−3 eV/K,

for 1.1 eV < E < 1.7 eV.

The reference curve for the absorption coefficient of silicon used here was obtained by interpolating the optical data of silicon [63,94]. The derivative dα(T )/dT is then calculated by dα(T ) dT

=

α(E + ∆E , T2 ) − α(E , T1 ) . T2 − T1

(4.18)

The results of the three different models for the TCA of silicon are shown in Fig. 54. Fig. 54 indicates clearly the restricted validity of model I, which shows agreement with model III only in the wavelength range around 1000 nm. A significant difference of models II and III is also evident. A comparison of our experimental data with the temperature coefficients of the spectral responsivity calculated with the three models is presented below. The influence of the different parameters determining the internal quantum efficiency ηi (λ) on the temperature dependence of the spectral responsivity was investigated using the temperature coefficient of model III for the absorption coefficient of silicon. Model III was used because it is the only model covering the whole wavelength region. In the first instance, the spectral response of a silicon trap detector was measured at several laser lines, and the resulting values were fitted with Eq. (4.10) [63]. The parameters obtained for Pf , Pb , t, d, h, and r are given in Table 4.3.

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Table 4.3 Parameters obtained by fitting the spectral responsivity vs. wavelength curve of a silicon trap detector according Eq. (4.10) Parameter

Fitted value

Pf Pb t D H r

0.9936 0.9985 0.82 µm 9.21 µm 260 µm 0.63

These values were taken as reference values for the theoretical investigations.

Table 4.4 Summary of the main influence parameters of the TCA categorised to wavelength ranges Wavelength range (nm)

Main influence parameter on TCA

<700

Pf , t, d Pb h, r

500–900 <900

Then, the values of Table 4.3 were taken as reference values, and only one of them was varied in turn to investigate its influence on the temperature coefficient, while all the others were kept fixed. The results are shown in Fig. 55(a)–(f). As expected, the parameters Pf , t, and d were found to be the factors that mainly influenced the temperature dependence in the short wavelength range up to 700 nm, while Pb is effective mainly in the mid wavelength region of 500–900 nm. h and r are responsible for the long wavelength region around 1000 nm (see Table 4.4). Considering the magnitude of the resulting changes, variations of parameters t and d, which determine the position of the pn junction and the end of the depletion region, do not have a significant influence on the spectral responsivity, in contrast to Pf and Pb , which determine the collection efficiency at the SiO2 /Si interface and in the bulk silicon have a large impact. At photon energies near the band gap, the thickness h of the diode and the reflection coefficient r of the radiation at the backside of the diode has a strong effect. This is attributed to the fact that photons with energies near the band gap have large penetration depths of the order of the thickness of the diode. To minimise the influence of h and r, it is desirable to have diodes with a thickness larger than the penetration depth of the photons and with high reflection coefficients at the backside. The temperature dependence of the spectral responsivity at normal room temperature was measured using a cryogenic radiometer set-up described in Ref. [68]. The trap detector was covered with a water jacket and the temperature of the water was controlled by a thermostat. The temperature of the aluminium block carrying the diodes was measured using a platinum resistor PT100. Investigations were performed at several fixed laser wavelengths (476, 488, 514, 568, and 647 nm) and at several wavelengths generated with a continuously tuneable Titanium-sapphire laser (900, 949, 984, 1000.2, and 1014.2 nm). The measured temperature coefficient of the spectral responsivity is depicted in Fig. 56. The relative change of the spectral responsivity with temperature was found to be smaller than 2 × 10−5 K−1 below 900 nm. At longer wavelengths, the temperature coefficient increases considerably, reaching values as high as 4.5 × 10−3 K−1 at 1014.2 nm. Combining the models for the temperature-dependent absorption coefficient α(T ) of silicon, the temperature dependence of the index of refraction of air and the temperature dependence of the reflectivity together with Eqs. (4.8) and (4.11), the temperature dependence of the spectral responsivity of the silicon trap can be predicted (Fig. 56). It can be seen that the Model III results are in excellent agreement with the experiment in the wavelength range above 900 nm, while model I shows significant deviation with increasing wavelength and is only able to approximately predict the trend of the experimental data. The results of model II show reasonable agreement with those of model III. However, the model III theory [93] is supposed to fit our experimental data over the whole wavelength range with good accuracy. Fig. 56 shows the temperature coefficient of the spectral responsivity in the shorter wavelength range on an extended scale. This shows that model III is not a sufficiently good fit for all wavelength ranges. It is clear that the theory has to include the temperature dependence of the trap reflectivity to predict the measured temperature dependence of the spectral responsivity for wavelengths shorter than 650 nm. Due to the lack of experimental data, only the values for 476 nm and 633 nm could be calculated. When the above is taken into account good agreement with the experimental results is found (model III). The remaining small deviation is probably due to the simplifying assumptions that the temperature dependence of reflectivity is independent of the angle of incidence and that α(T ) is the only temperature-dependent parameter of ηi (λ). However, as the overall relative uncertainty in the measurement of spectral responsivity [63] is 1 part in 104 , uncertainties in the temperature of the photodiodes of the order of 1 K do not influence the overall accuracy for wavelengths smaller than 900 nm. At longer wavelengths, temperature has to be controlled much more strictly. For longer wavelengths, the value of the temperature coefficient of the reflectivity is available only at 1047 nm. Comparing this value (given in Table 4.2) and the measured temperature coefficient of the spectral response shows that the reflectivity change has almost negligible influence on the overall temperature coefficient. This is confirmed by the excellent agreement between theory and experiment in the long wavelength region, confirming the assumption that the trap reflectivity can be neglected. In this region the deviation

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Fig. 55. Temperature coefficient of the spectral responsivity as a function of wavelength. A set of reference parameters were obtained by fitting the measured spectral responsivity curve of a silicon trap detector. For each graph, all parameters except the indicated one, were kept constant at their reference value from Table 4.3 [125] (courtesy Optical Society of America).

of theory and experiment is less than 10%, and the theory can be used for correcting experimentally obtained data with high accuracy. The model describes the temperature coefficient of the spectral responsivity of silicon photodiodes and trap detectors as a function of wavelength. The wavelength-dependent temperature coefficient is well described by the temperaturedependent absorption coefficient of silicon in connection with the analytical formula for the internal quantum efficiency. Comparative measurements of the temperature dependence of the spectral responsivity for silicon trap detectors in the wavelength range from 476 to 1015 nm at room temperature have been performed. The predictions of the model are in good agreement with the measured values. The obtained analytical theory of the temperature coefficient of the spectral responsivity is an essential tool for further increasing the accuracy of spectral responsivity scales based on silicon photodiodes, especially in the near infrared region beyond 1000 nm. 5. Transfer of the spectral responsivity of the reference detector to the filter radiometer The radiometric determination of thermodynamic temperatures is performed by irradiance measurements with absolute calibrated narrow-band interference filter radiometers (FRs). The calibration of the FRs is a two-step process. First, a broadband transfer detector (Si trap detector for UV and VIS or single InGaAs diode for NIR) is calibrated against a cryogenic radiometer at several discrete laser emission lines as

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Fig. 56. Measured relative change in spectral responsivity (open circles) of a silicon trap detector as a function of wavelength together with theoretical predictions: Prediction using the various models for the temperature dependence for the absorption coefficient of silicon: solid squares model II [92], solid circles model I [91], line model III [93], crosses model III together with the temperature coefficient of the Trap reflectivity RTrap [125] (courtesy Optical Society of America).

described in Section 4. For silicon photodiodes the data is then fitted with the formula given in Eq. (4.4). The transfer detector is then used to continuously calibrate the FRs at a spectral comparator facility. By applying a tuneable Ti:sapphire ring laser, the capabilities for the calibration of silicon trap detectors has been extended up to 1015 nm [63]. This made it possible to calibrate silicon photodiode-based FRs with centre wavelengths up to 1015 nm for the determination of thermodynamic temperatures. Based on the application of two diode lasers (1.31 and 1.55 µm) and an interpolation with bolometers the calibration of InGaAs diode transfer detectors has been extended to 1.6 µm [74]. Following the design of the silicon photodiode FRs, a novel FR applying an InGaAs photodiode has been constructed with a centre wavelength at 1.595 µm and a bandwidth of about 100 nm. To ensure an accurate calibration of these FRs against the transfer standards the intercomparison procedure needed to be improved, because special problems such as polarization, stray light and temperature dependence of the detectors significantly affect the calibration at longer wavelengths.

5.1. Experimental set-up

For the calibration of the FRs against the transfer detectors, the spectral comparator facility described in Ref. [67] was improved. The new setup is shown in Fig. 57. The calibration of narrowband detectors against broadband transfer detectors requires that the out-of-band stray light contribution to the signal must be suppressed to 10−6 . A more detailed view on the effect of stray light on the spectral responsivity measurement is given in Section 5.6. Typical values for the stray light contribution of single grating monochromators are 10−5 . To achieve the required lower stray light level we incorporated a prism monochromator (flint or suprasil prism) as a predisperser. The combination of a prism monochromator with a grating monochromator has two advantages over a double-grating monochromator, in particular the transmitted flux is higher and higher-order suppressing filters are not required. For the calibration of the silicon diode-based narrow-band FRs up to 1200 nm we use gratings of 1302 lines mm−1 and 1200 lines mm−1 , resulting in a spectral dispersion of about 2 nm mm−1 . The calibration of InGaAs FRs is performed with gratings of 651 lines mm−1 and 325.5 lines mm−1 , providing a spectral dispersion of 4 nm mm−1 and 8 nm mm−1 . The spectral bandwidth of the pre-disperser is set to values between 20 and 45 nm. The wavelength setting of both monochromators has been calibrated at about 50 spectral emission lines of different spectral lamps. The grating and the prism are driven synchronously in wavelength by two stepper motors. Behind the exit slit of the grating monochromator the radiation is collimated. To consider the influence of polarization on the spectral responsivity of the detectors, a Glan–Thompson polarization prism is inserted in the beam behind the grating monochromator. The polarizer is slightly tilted with respect to the optical axis to avoid interreflection with the apertures of the detectors. A detailed view on the aspects of polarization is given in Section 5.5. The calibration procedure, apart from the consideration of polarization, is explained in detail in Ref. [67]. An overview of the spectral responsivity of the filter radiometers presently used at PTB is given in Fig. 58.

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Fig. 57. Improved spectral comparator facility for the calibration of narrow-band filter radiometers against broadband transfer detectors [106].

Fig. 58. Overview of the spectral responsivity of the narrow-band interference filter radiometers (silicon and InGaAs-based) presently used at PTB for the determination of thermodynamic temperature. The grey lines show the spectral responsivity of the used silicon and InGaAs photodiodes [106].

5.2. Temperature dependence of the FRs Applying filter radiometers for thermodynamic temperature measurements of blackbody radiation below 500 ◦ C requires centre wavelengths above 900 nm. The temperature dependence of the responsivity of silicon photodiodes increases rapidly at wavelengths above 900 nm, reaching values of 4.5 × 10−3 K−1 at 1014.5 nm as described in Section 4. Additionally, the centre wavelength of the interference filters changes with temperature. We performed calibrations of the spectral responsivity at three different temperatures in the range from 22 to 27 ◦ C for several filter radiometers (centre wavelength 800 nm, 900 nm, 1000 nm and 1.595 µm) while keeping the temperature of the reference detector constant. The relative change of the integral spectral responsivity and the absolute shift of the centre wavelength with temperature are shown in Table 5.1. A drift of the centre wavelength towards longer wavelengths of about 30 pm K−1 for the Si filter radiometers (15 nm FWHM) and 85 pm K−1 for the InGaAs FRs (100 nm FWHM) is observed. The change of the integral spectral responsivity with temperature is a combined effect of the temperature dependence of the interference filter and the detector. For the 800 nm FR and the 900 nm FR, where the temperature coefficient of the Hamamatsu S1337 is negligible, the influence of the interference filter is dominant. At 1000 nm for the Si FR and at 1.595 µm for the InGaAs FR the temperature coefficient of

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Table 5.1 Temperature coefficient of the filter radiometers

λc (nm)

(∆Is /Is )(1/∆T ) (K−1 )

(∆λc /∆T ) (pm K−1 )

800 900 1000 1595

−2.8 × 10−4 0.2 × 10−4 13.5 × 10−4 9.1 × 10−4

+33 +30 +35 +85

the detector becomes dominant, resulting in an increase of spectral responsivity with temperature. The temperature of the detectors during their calibration and application is controlled within ±50 mK by a Pt100 sensor to ensure that the effect of temperature on the integral spectral responsivity Is is less than 1 × 10−4 . 5.3. Non-linearity of the spectral responsivity Non-linearity measurements of the spectral responsivity of silicon photodiodes (Hamamatsu S1337) for wavelengths up to 799 nm showed that these diodes are linear within 1 × 10−4 for photocurrents below 0.3 mA [77]. Recent investigation of their spectral responsivity at wavelengths of 900 and 1000 nm revealed a strong supralinearity (see Fig. 48). This requires a correction of the spectral responsivity of the trap detector determined with the Radiation Thermometry Cryogenic Radiometer (RTCR), when used at the spectral comparator. The photocurrent of the trap detector during its calibration is of the order of 350 µA. When the trap detector is used for the calibration of filter radiometers with the spectral comparator facility the photocurrent of the trap is about 150 pA at 900 nm and 420 pA at 1000 nm. Applying the theory outlined in Section 4, the resulting correction to the spectral responsivity of the trap detector is of order 7 × 10−4 at 900 nm and 2 × 10−3 at 1000 nm. 5.4. Polarisation effects Due to the use of reflecting surfaces the radiation passing through the comparator facility is polarized. The degree of polarization behind the exit slit is a function of wavelength and depends on the applied grating and prism. To investigate the magnitude of the effect of polarization, the 1000 nm filter radiometer was calibrated in two perpendicular positions rotated on the optical axis without a polarizer inserted in the optical path. The relative difference in the integral spectral responsivity between the two perpendicular positions was 3 × 10−3 . For thermodynamic temperature measurements, the filter radiometers are used in front of the PTB blackbodies, which are unpolarized sources. Thus, for our applications the spectral responsivity of the filter radiometers for unpolarized light has to be calibrated. To achieve this, a Glan–Thompson prism was placed behind the exit slit of the monochromator. With this setup we calibrated the FR in two orthogonal positions, while keeping the prism polarizer and the transfer detector fixed. Using the Stokes–Mueller formalism [95], the spectral responsivity for unpolarized light is the arithmetic mean of the two orthogonal measurements. This method has two advantages. First, the trap detector is used with light in the same polarization state as calibrated at the RTCR. It avoids a possible polarization effect of spectral responsivity for non-ideal trap detectors as described in Ref. [96]. Second, the calibration of the FR is independent of the polarization generated by the monochromator. 5.5. Stray light It is well known that monochromators transmit radiant flux outside their nominal bandpass, generally referred to as ‘stray light’. When calibrating a narrow-band detector against a broadband detector the level of stray light has a significant influence on the spectral responsivity. We have calibrated the 900 nm filter radiometer using two different gratings, with and without using the prism monochromator as predisperser. The slit width of the monochromator was adapted, resulting in the same spectral bandwidth of 0.5 nm for all gratings used. To exclude the grating-dependent influence of polarization, two calibrations in orthogonal positions with linear polarized radiation were performed. Without the predisperser the normalized integral spectral responsivity for unpolarized radiation revealed a significant difference of 2 × 10−3 (left-hand side of Fig. 59) for the two gratings. This difference can be explained by the presence of stray light in combination with the different spectral responsivities of the filter radiometer and the transfer detector. The photocurrent of an arbitrary detector when used in the calibration procedure at the spectral comparator is the product of three wavelength-dependent factors: the spectral flux of the source, the transfer function of the monochromator and the spectral responsivity of the detector. For an ideal monochromator the transfer function is zero outside its bandpass. For a non-ideal monochromator the transfer function decreases to a non-zero value, the stray light. In the case of a broadband reference source, the photocurrent of the detector is composed of two contributions, one from the bandpass and, due to a non-zero out-of-bandpass transfer function, one from stray light. In the calibration of FRs two cases can be distinguished. First, if the set wavelength of the monochromator is within the bandpass of the FR, the determined spectral responsivity for the latter will be too low. This is due to the fact that the photocurrent of the trap is higher, as the spectral responsivity of the trap is not negligible in the wavelength interval where the stray light occurs, while the spectral responsivity of the FR is negligible in this wavelength

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Fig. 59. Effect of stray light on the integral spectral responsivity of the 900 nm FR. The bars denote the standard uncertainty of the calibration [106].

range. Second, if the set wavelength is outside of the bandpass of the filter radiometer, the resulting spectral responsivity will be too high, because the main contribution of the photocurrent of the FR being from the stray light region, i.e. from the bandpass [67]. Incorporation of the prism monochromator in front of the grating monochromator is equivalent to limiting the wavelength interval of the source flux. As a result the stray light contributions will be minimized. An experimental proof for this is shown in Fig. 59 (right-hand side). Even though the prism monochromator introduces a stronger polarization effect, the combined result of the two orthogonal measurements for the integral of the spectral responsivity is the same for both gratings within the standard uncertainty of the calibration. 5.6. Uncertainty of the filter radiometer calibration The uncertainty budgets of the irradiance measurements with the FRs are given in Table 5.2. The contributing uncertainty of the spectral responsivity of the transfer detectors is explained in detail in Section 4 for Si and in Ref. [74] for InGaAs. The areas of the apertures of the transfer detectors were measured with the procedure outlined in Section 3; the absolute uncertainty of the diameter is below 0.4 µm. Due to its smaller sensitive area, a 3 mm diameter instead of the generally used 5 mm diameter aperture was used in front of the InGaAs transfer detector, resulting in a larger contribution to the uncertainty of the aperture area. In contrast to the calibration at the RTCR, at the spectral comparator the apertures of the transfer detectors are overfilled and diffraction losses occur, which have to be corrected. The collimated beam of the monochromator exhibits a slight divergence of about 0.2◦ . Therefore, the spectral irradiance is a function of the distance of the aperture plane from the exit slit. The alignment error of the two aperture planes with respect to one common plane is 0.02 mm, leading to the uncertainty contribution from the variation of the spectral irradiance. The uncertainty resulting from wavelength-dependent effects, like the temperature coefficient of the interference filter, the calibration uncertainty of the monochromator and the reproducibility of the centre wavelength after a new alignment, is calculated for the lowest blackbody temperature (value in parenthesis given in Table 5.2) at which each filter radiometer can be used for reasons related to the signal-to-noise ratio. The contribution of the temperature coefficient is calculated on the basis of a 50 mK uncertainty in the temperature control of the detectors. The described, improved technique for the calibration of filter radiometers in the wavelength range from 400 nm to 1.6 µm at the spectral comparator facility of the PTB minimizes by complementing the existing monochromator with a prism predisperser, the influence of stray light. By using a polarizer and performing the calibration in two orthogonal positions, filter radiometers can be calibrated for use with unpolarized sources. The influence of temperature on the interference filters and the photodiodes is considered and significantly reduced by a careful control of the temperature of the detectors. With this calibration method narrow-band filter radiometers can be calibrated with uncertainties lower than 3 × 10−4 between 400 nm up to 900 nm and about 6 × 10−4 up to 1000 nm. For the InGaAs filter radiometer the overall uncertainty is 2 × 10−3 , the main contribution resulting from the InGaAs transfer detector. 6. Development of new high-temperature fixed-points above 1400 K 6.1. Metal-carbon eutectics As mentioned in the introduction, accurate reference sources are required for photometry, radiometry and thermometry. In addition, because of rapidly rising energy costs and demands for reductions in environmental pollution, industry is being

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Table 5.2 Contributions to the relative standard uncertainty (k = 1) of the measurement of the spectral irradiance in front of a blackbody for the different filter radiometers Filter radiometer

∆/s × 104 676 nm

800 nm

900 nm

1000 nm

1595 nm

1. Spectral responsivity of transfer detector 2. Non-linearity correction to transfer detector 3. Aperture area of transfer detector 4. Diffraction of transfer detector aperture 5. Distance from exit slit 6. Temperature coefficient of transfer detector 7. Temperature coefficient of FR 8. Homogeneity of spectral comparator beam 9. Stability of Tungsten–halogen lamp 10. Reproducibility 11. Uncertainty of centre wavelength (8 pm, 1595 nm: 35 pm) Sum in quadrature ×104 (stated in [3]) In terms of temperature uncertainty (mk)

1.0 – 1.6 1.0 0.3 0.1 0.5 0.2 0.2 0.5 1.7 (800 ◦ C)

1.0 – 1.6 1.0 0.3 0.1 0.4 0.2 0.2 0.5 1.4 (660 ◦ C)

1.0 1.0 1.6 1.0 0.3 0.1 0.3 0.2 0.2 0.5 1.5 (457 ◦ C)

4.4 1.0 1.6 1.0 0.3 2.0 0.5 0.2 0.2 0.5 1.3 (419 ◦ C)

17.0 – 2.7 1.0 0.5 – 0.3 0.2 0.2 0.5 1.8 (419 ◦ C)

2.9 (3.5 in [3]) 16 (800 ◦ C)

2.7 (3.7 in [3]) 13 (660 ◦ C)

2.9 10 (457 ◦ C)

5.5 18 (419 ◦ C)

17.4 93 (419 ◦ C)

Table 6.1 List of possible metal-carbon and metalcarbide-carbon eutectics Material

Melting temperature (K)

Fe–C Co–C Ni–C Pd–C Rh–C Pt–C Ru–C Ir–C Re–C MoC–C TiC–C ZrC–C HfC–C

1426 1597 1602 1765 1930 2011 2227 2564 2747 2856 3034 3155 3458

forced to accurately control their processes. Thus requires the exact knowledge of the parameters involved, one of the most important of which is temperature. In contrast to low and medium temperature measurements, which are usually performed with contact thermometers, high-temperature measurements, especially in the temperature range above 1400 K, are usually performed by non-contact radiation thermometry. For non-contact radiation thermometry the emitted thermal radiation from the hot body is measured and the temperature calculated using Planck’s law of thermal radiation. To calibrate radiation thermometers radiation sources with accurately known thermal radiation are needed. Unfortunately, the demand for high accurate temperature measurements at elevated temperatures above 1400 K and the availability of highly accurate thermal radiation sources at high temperatures, have been hindered due to the lack of stable and accurate high-temperature fixed-points. Recently, a new family of high-temperature fixed-point thermal radiation standards based on metal-carbon and metal carbide-carbon (M(C)–C) eutectic have been developed. This class of materials offers highly reproducible phase transition temperatures ranging from 1400 K up to 3300 K and even higher, reproducible within a few tenth of a kelvin [29]. For a list of possible M(C)–C eutectics see Table 6.1. These materials, for the first time, opened a new way for practical and accurate high-temperature standard radiation sources. The practicability and the reproducibility of these novel materials as temperature fixed-points has been the subject of an European funded project entitled Novel High-Temperature Metal-Carbon Eutectic Fixed-Points for Radiation thermometry, Radiometry, and Thermocouples (HIMERT) [97]. This project has allowed Europe to develop a substantial knowledge base in this important high-temperature technology. An overview of the scientific work done in the field of M(C)–C eutectics up to the year 2006 is given in Ref. [98]. Normal fixed-points are based on a pure element, e.g. silver, gold or copper, in a graphite crucible. These are not practical at higher temperatures because graphite dissolved from the crucible into the fixed-point metal, depressing the fixed-point temperature. Typical examples are the platinum and the palladium fixed-points, which can be realised with an uncertainty of only 0.4 K [99]. However, for these new materials the effect has been turned to advantage through utilizing the eutectic point of the phase diagram of pure metal and carbon. The crucible made of carbon can no longer contaminate the metal as carbon is one of the fixed-point materials. In particular the M(C)–C eutectics the graphite material of the crucible is part of the fixed-point material and will only change the graphite content of the fixed-point, but will have no effect on the fixed-point temperature only on the duration

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Fig. 60. Phase diagram of a typical eutectic and resulting freezing curve when going along the indicated line.

Fig. 61. Typical melting and freezing plateau of Re–C.

of the fixed-point plateau. This can be understood while looking at the phase diagram of a typical eutectic shown in Fig. 60. Cooling the melt along the red line will be seen as a linear cooling in the temperature versus time curve. As soon as the liquidus line is crossed the excess carbon will freeze out, changing the carbon content in the melt, resulting in a movement along the liquidus line towards the eutectic point. This effect results in a change in the slope in the freezing curve shown in Fig. 60. When the eutectic concentration is reached the eutectic will freeze out at a constant temperature, yielding the freezing plateau. This plateau will last until all the eutectic mixture is frozen. Finally, the frozen eutectic will cool down linearly. For a proper operation of a eutectic mixture, not affecting its robustness, it is essential that the graphite content of the mixture is slightly higher than the eutectic concentration. The whole process will also work if the graphite content is smaller than the eutectic concentration. However, then additional graphite will be dissolved out of the crucible material, potentially reducing the mechanical stability of such a cell. Keeping the graphite concentration higher than the eutectic concentration, the eutectic reaction makes the fixed-point highly reproducible and insensitive to crucible contamination effects. However, filling is done with slightly metal-rich mixtures, this allows the formation of a better ingot, due to the lower viscosity of the metal-rich eutectic mixture. The effect of dissolving the required graphite from the crucible, potentially weakening it, is reduced by introducing an inner sacrificed sleeve [98]. In contrast to a one-element fixed-point, the frozen eutectic mixture does not have a homogeneous structure but is made of two different phases: the metallic and the graphite phase. The separation of these phases is eventually determined by the freezing rate. Therefore, the melting point is potentially influenced by the cooling rate of the previous freeze and the thermal annealing procedure [100]. Fortunately, for most M(C)–C eutectics the melting temperature is only slightly or even not influenced by the thermal history of the cell [100]. In consequence, the melting plateau is used as temperature fixed-point and not the freezing plateau as it is the case for the one element fixed-point of the present high-temperature scale. A typical melt and freezing plateau is shown in Fig. 61. As can be drawn from Fig. 61 the melting plateau is rather sloped. Therefore, a distinct melting point temperature can not be defined by taking the average temperature of the melting plateau as it is done in case of the conventional one-element fixed-points. For comparing the melting behaviour of eutectic fixed-point cells the point of inflection of the melting curve

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Fig. 62. Melting plateau of a Pt–C eutectic observed during different heating rates (Left side overview, right side detail in the vicinity of the point of inflection).

Fig. 63. Typical design for a radiance eutectic fixed-point cell.

is the currently agreed method for identifying the melting point of the respective eutectic. If the freezing temperature is determined the highest temperature during the freezing plateau is taken as freezing temperature. In fact, the main advantage of the point of inflection is its high reproducibility (much better than 100 mK) and its relative insensitivity to the heating rate, as can be seen in Fig. 62. It can be seen from Fig. 62 that the temperature of the point of inflection is nearly independent on the heating rate. 6.2. Fixed-points made from metal (carbide) carbon eutectics To make use of the high phase-transition temperatures of the metal(carbide) carbon eutectics, special fixed-point designs have been developed. Within the EU funded project HIMERT the typical design depicted in Fig. 63 has been used. The cells in Fig. 63 have an overall length of about 45 mm and an outer diameter of about 25 mm. The radiating cavity has 3 mm in diameter and is about 35 mm long, resulting in an emissivity of about 0.9995. Due to the small aperture diameter such cells can only be observed with radiation thermometers having imaging optics, those designed for measuring spectral radiance. Such cells are therefore called radiance mode or simply radiance cells. For the investigation with filter radiometers having no imaging optics, cells with a large diameter of the radiating aperture have to be developed. Investigating such cells with filter radiometers, the spectral irradiance is measured. Consequently such cells are called irradiance mode or simply irradiance cells. Cells with a diameter of 10 mm have been developed by the All Russian Institute of Applied Optics (VNIIOFI) and are shown in Fig. 64 [101]. More details about the manufacturing and optimization of the eutectic fixed-point cell design can be found in [102,103]. Using the radiance cells depicted in Fig. 63 the repeatability and the reproducibility of the M(C)–C phase transition has been investigated. Measurement of the absolute temperature of the M(C)–C phase transitions has been determined using both radiance and irradiance mode cells.

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Fig. 64. Typical design for an irradiance eutectic fixed-point cell developed by the VNIIOFI [101].

Fig. 65. Schematic of the experimental set-up for determining the reproducibility of the MC eutectics [50].

6.3. Investigation and temperature determination of metal(carbide) carbon eutectics Before novel high-temperature fixed-points can be implemented in a new International Temperature Scale, or can serve as a standard source of calculable spectral radiation, their reproducibility must be established and their absolute melting or freezing temperatures must be determined. Reproducibility in this case means that M(C)–C eutectics of the same type but manufactured by different institutes or companies with materials from different suppliers shall give the same melting or freezing temperature within ±100 mK above 2300 K. To assess the reproducibility of the cells no absolute temperature measurements are necessary but a relative comparison is sufficient. The experimental setup for determining the reproducibility is shown in Fig. 65. In this investigation the fixed-points under evaluation are placed in two quasi-identical furnaces supplied by the NMIJ and their spectral radiances were compared with two different radiation thermometers, one from the LNE-INM/Cnam and the other from PTB. In total 15 different cells were investigated. These were manufactured by LNE-INM/Cnam, NMIJ, and NPL. Cells made from five different materials were measured, namely Co–C, Pd–C, Pt–C, Ru–C, and Re–C. The spectral radiance L1 of one cell during its melting was recorded by the radiation thermometers and then compared with the spectral radiance L2 recorded on the other cell, made from the same material but from a different manufacturer. The relative difference in radiance (L1 − L2 )/L = ∆L/L can then be converted in a temperature difference ∆T by the following formula [104].

∆T =

∆L λ T 2 L

c2

.

(6.1)

During the experiments the repeatability, i.e. the variation of the temperature of the point of inflection of one single cell was also assessed. The results of these experiments are shown in Fig. 66. The reproducibility between different cells from different manufacturer is shown by the scatter of the date for one material in Fig. 66. It can be seen in Fig. 66 that the reproducibility is better than ±100 mK for Co–C, Pt–C, and Re–C and better than ±200 mK for Pd–C and Ru–C. The repeatability of the single cells, indicated by the error bars assigned to the symbols in Fig. 66, is in every case better than 100 mK. Note that these error bars also include the stability of the radiation thermometers, which means that the repeatability of only one cell will be significantly better than 100 mK.

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Fig. 66. Results of the repeatability and reproducibility measurements [50].

Fig. 67. Set-up of the absolute temperature measurements of the eutectics.

Now the repeatability and the reproducibility of the cells has been determined, the next step is the absolute temperature determination of the eutectics. For these measurements the absolute spectral irradiance has to be measured with an uncertainty smaller than the radiation thermometric measurement according the ITS-90. For example for the material combination with the highest melting temperature, i.e. Re–C with 2747 K, the uncertainty of the temperature determined according the ITS-90 is 200 mK at a coverage factor k = 1. This is the theoretical limit obtainable using the ITS-90. However, in any case a radiation thermometer has to be used, which increases the practically obtainable uncertainty by a factor of 5–10. Using the formula given in Eq. (6.1), results in a relative uncertainty of the measured spectral irradiance of 5.6 × 10−4 for the theoretical limit of the ITS-90 of 200 mK. The challenge is therefore to determine spectral irradiance for these cells with an uncertainty smaller than 5.6 × 10−4 . To perform the absolute temperature determination the apparatus was setup as in the schematic diagram shown in Fig. 67. As the diameter of the aperture of the eutectic fixed-points is only 3 mm and therefore too small for absolute radiometric measurements without imaging optics the following method has been developed. The thermodynamic temperature of a high-temperature blackbody (HTBB) was measured using absolutely calibrated filter radiometers in the irradiance mode at a temperature close to the melting temperature of the high-temperature fixed-point under test. Subsequently a radiation thermometer of type LP3 has been used for a relative comparison of the spectral radiance of the HTBB and the eutectic fixedpoint under investigation. Knowing the absolute temperature of the HTBB and the difference in spectral radiance between HTBB and eutectic fixed-point the absolute temperature of the eutectic fixed-point can be calculated according Eq. (6.1). The resulting temperatures are presented in Table 6.2. The results obtained on the eutectics manufactured by NPL have been compared with the results obtained by NIST half a year before on the same cells. The difference of these measurements together with their combined uncertainty are shown in Fig. 68. It can be deduced from Fig. 68 that the agreement between the NIST and the PTB thermodynamic temperature measurements is better than their combined uncertainty. The standard uncertainty of the PTB results is smaller than 200 mK.

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Table 6.2 Thermodynamic temperatures of the M(C)–C fixed-point obtained with a measurement according Fig. 67 Material

Manufacturer

Melting temperature (K)

Uncertainty (k = 2)

Ru–C Ru–C Ru–C

NMIJ NPL BNM

2227.44 2227.22 2226.96

0.40 0.39 0.37

Pt–C Pt–C Pt–C

NMIJ NPL BNM

2011.88 2011.74 2011.74

0.33 0.32 0.30

Pd–C Pd–C Pd–C

NMIJ NPL BNM

1765.19 1765.07 1765.00

0.26 0.27 0.26

Co–C Co–C Co–C

NMIJ NPL BNM

1597.30 1597.16 1597.23

0.23 0.22 0.21

Fig. 68. Difference of the absolute temperatures of the NPL cells obtained by PTB and NIST. The dotted line shows the combined standard uncertainty.

Fig. 69. Schematic diagram of the blackbody configuration, showing the eutectic fixed-point cell, the stray light baffles and the precision aperture on the left and the two types of detectors on the right.

The radiating areas of 3 mm in diameter of the M(C)–C cells investigated so far have been too small for direct application in radiometry and photometry. For that purpose the cells shown in Fig. 64 have been developed by the VNIIOFI. Using cells with a radiating area of 10 mm in diameter allows the position of a precise aperture in front of the furnace, defining the exact radiating area according Fig. 69. The precision aperture, defining the radiating area of the fixed-point is positioned outside the HTBB cavity, to enable a good temperature stabilisation. Due to the large distance between the aperture and the fixed-point cell, the precision aperture has to be as small as 3 mm in diameter, to view only the bottom of the radiating fixed-point cell. The large distance between the radiating cell and the precision aperture requires the use of stray light baffles. These become even more important for the additional radiance temperature measurement when the radiation thermometer LP3 is used. For these radiance temperature measurements the precision aperture is removed from in front of the blackbody and the LP3

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Table 6.3 Standard uncertainty (k = 1) of the ITS-90 temperature measured with the radiation thermometer TSP and the LP3 T ( K)

u(TSP ) (K)

u(LP3) (K)

2800 3000 3200

0.60 0.75 0.80

0.75 0.86 0.98

Table 6.4 Standard uncertainty (k = 1) of the thermodynamic temperature measured with the filter radiometers T ( K)

u(FD17) (K)

u(FR 676) (K)

u(FR 800) (K)

u(FR 900) (K)

2800 3000 3200

0.32 0.36 0.41

0.39 0.45 0.50

0.46 0.53 0.59

0.51 0.58 0.66

Fig. 70. Overview of the spectral responsivities of the applied filter radiometers [101].

is focused onto the entrance aperture of the fixed-point cell. The stray light baffles inside the HTBB, minimize the thermal radiation originating from outside the fixed-point cell and reaching the filter radiometer and the radiation thermometer. Using this method three eutectic fixed-point materials have been investigated. One metal-carbon eutectic, namely Re–C, with a melting and freezing temperature around 2747 K and two metal carbide-carbon eutectics, namely TiC–C and ZrC–C, with melting and freezing temperatures around 3030 K and 3150 K, respectively [105]. Two types of detectors were used for the determination of the melting and freezing temperatures. Radiation thermometers for measuring temperatures according to the International Temperature Scale of 1990 (ITS-90) and filter radiometers for measuring thermodynamic temperatures. The radiation thermometers used, a TSP from the VNIIOFI and an LP3 from the PTB (see Fig. 19) have been calibrated traceable to the ITS-90. The standard uncertainty of the temperature measurement using the TSP and the LP3 is given in Table 6.3. The spectral irradiance responsivity of the filter radiometers from the PTB have been calibrated traceable to the cryogenic radiometer of the PTB as described in Section 4 [67,106], yielding thermodynamic temperatures, independent on the ITS90. Two different types of filter radiometers were used: Three narrow-band interference filter radiometers with centre wavelengths around 676, 800, and 900 nm, and one broad-band glass filter radiometer, having a centre wavelength around 550 nm [107]. The standard uncertainty of the thermodynamic temperatures measured with the filter radiometers, including the uncertainty in the spectral responsivity of the filter radiometers, ranged from 5 × 10−4 up to 2.5 × 10−3 . Due to the large radiating aperture of the irradiance mode cells the temperature drop at the bottom due to radiative cooling has to be taken into account, and have to be determined by numerical modelling [108]. This temperature drop has been measured as 70 mK for the ZrC–C fixed-point cell used here [105]. The measured values were not corrected for this temperature drop, but the value has been included in the uncertainty budget, because the measuring geometry was different. The overall uncertainty, including also the uncertainty of the diffraction correction, and the uncertainty due to the geometry, is given in Table 6.4. The spectral responsivities of the filter radiometers used in these measurements are summarized in Fig. 70. In a first step, the TiC–C and ZrC–C eutectic fixed-point cells were measured at VNIIOFI using the broad-band filter radiometer FD17 from PTB and the TSP radiation thermometer from VNIIOFI. Then the TiC–C and ZrC–C cells, and additionally Re–C eutectic fixed-point cells, were measured at PTB using the broad-band filter radiometer, three narrow-

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Fig. 71. Typical melting curves of TiC–C, ZrC–C, and Re–C [101]. Table 6.5 Temperature correction in kelvin due to diffraction effects for the measurements at the PTB

Re–C TiC–C ZrC–C

FD17

FR 676

FR 800

FR 900

0.48 0.55 0.63

0.65 0.75 0.85

0.77 0.89 1.01

0.87 1.00 1.14

Table 6.6a Temperatures in kelvin for the measurements at VNIIOFI TiC–C

FD 17

TSP

Average ± u Std. dev.

3031.6 ± 0.36 –

3032.17 ± 0.75 –

ZrC–C Average ± u Std. dev.

FD 17 3154.28 ± 0.41 –

TSP 3154.48 ± 0.80 –

band interference filter radiometers and an LP3 radiation thermometer from PTB. A schematic diagram of the experimental set-up used at VNIIOFI and PTB is given in Fig. 69. The precision aperture in front of the HTBB has a diameter of 3 mm, while the apertures of the filter radiometers are approx. 5 mm in diameter, the distance between the two apertures was about 1110 mm. For the measurements at VNIIOFI and PTB the same 3 mm aperture was used. During the measurements with the radiation thermometers this aperture was removed from in front of the HTBB. After heating the HTBB, the furnace was stabilized at a temperature approx. 25 K below the melting temperature of the eutectic material. The melting process was initiated by a step-like current increase of approx. 40 A after which the furnace was stabilized at a temperature of about 25 K above the melting temperature. For the freezing of the eutectic alloy the temperature in the furnace was reduced and stabilized at a temperature of about 25 K below the freezing temperature. Altogether more than 100 melting/freezing cycles were measured. Typical melting curves of the investigated eutectics are given in Fig. 71. From these profiles the point of inflection was taken as the transition temperature of the cell. For the filter radiometers the obtained temperatures were additionally corrected for diffraction losses, occurring at each of the two precision apertures. The overall correction due to this effect is given in Table 6.5. The results for each instrument are presented in Tables 6.6a and 6.6b. The standard deviation given in Tables 6.6a and 6.6b is an indication of the repeatability of the cells and it can be seen that in all, except three cases, the standard deviation is significantly smaller than the uncertainty of the thermodynamic temperature measurement showing the good repeatability of the irradiance mode fixed-point cells. A correction to the temperatures to the emissivity of ε = 0.9996 ± 0.0001 has been applied. The results presented in Tables 6.6a and 6.6b were the first thermodynamic temperature measurements of eutectic fixedpoint cells by absolute radiometry using filter radiometers in the irradiance mode. The results obtained at VNIIOFI and PTB all agree within their combined uncertainty, indicating that the reproducibility of the large irradiance cells in different furnaces measured with different detectors is very good. Especially all measurements using the filter radiometers are in very good agreement. The systematically higher temperatures measured with the radiation thermometers are partly attributed to additional stray light effects, originating from the stray-light baffles. Instead of knife edges, these baffles have a thick land of 5 mm, at which reflections of the radiation coming from the cavity may occur. This additionally reflected light is blocked by the 3 mm aperture used for the absolute radiometric measurements and therefore has less influence on the thermodynamic temperature measurements. As the 3 mm aperture is removed for the radiance measurements reflections from the stray

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Table 6.6b Temperatures in kelvin for the measurements at PTB Re–C

FD 17 (2x)

FR 676 (6x)

FR 800 (10x)

FR 900 (2x)

LP3 (17x)

Average ± u Std. dev.

2745.55 ± 0.32 0.45

2746.15 ± 0.39 0.72

2745.81 ± 0.46 0.33

2745.48 ± 0.51 0.25

2747.40 ± 0.75 0.49

TiC–C Average ± u Std. dev.

FD17 (5x) 3031.22 ± 0.36 0.16

FR 676 (11x) 3031.13 ± 0.45 0.22

FR 800 (6x) 3031.00 ± 0.53 0.14

FR 900 (1x) 3030.53 ± 0.58

LP3 (8x) 3031.78 ± 0.86 0.24

ZrC–C

FR 676 (3x)

FR 800 (6x)

LP3 (5x)

Average ± u Std. dev.

3153.86 ± 0.50 0.16

3153.78 ± 0.59 0.47

3154.80 ± 0.98 0.27

The number of measurements averaged is given in brackets.

light baffles might well significantly influence these measurements. Currently this effect is under investigation by using different stray light baffles. Beside this other effects as those described below might also influence the difference between thermodynamic and radiation thermometric temperature. Despite the good agreement of the obtained results the measured melting temperatures are significantly lower than previously obtained temperatures using the small radiance eutectic cells [109,110]. This may partly be explained by the large dimensions of the irradiance cells and a possible inhomogeneous temperature distribution inside the HTBB. It has already been described in literature that a non-uniform temperature distribution over the length of the eutectic fixed-point cells significantly effects the obtained melting and freezing plateau shapes [111]. As the irradiance cells are about a factor of two longer than the radiance cells they are more sensitive to temperature inhomogeneities inside the furnace. Due to the special design of the HTBB, which requires a total disassembly and subsequent reassembly in order to exchange the eutectic cells, the temperature distribution might have changed after reassembling, although the overall configuration has not been changed. Presently VNIIOFI is developing a new HTBB furnace with an inner cavity diameter of 57 mm, enabling an improved temperature homogeneity across the length of the cells [112]. Beside the temperature in-homogeneity across the large eutectic fixed-point cells additional effects, e.g. impurity of the eutectic, may also affect the measured temperatures. Additional measurements on the large eutectic fixed-point cells from VNIIOFI are planned at PTB to further investigate the thermodynamic temperature measurements of the metal carbon eutectic phase transition. The results show that the temperatures of the phase transition of eutectic fixed-point materials can be obtained by absolute irradiance measurements using filter radiometers with their spectral responsivity calibrated traceable to the cryogenic radiometer. This is an essential step towards an implementation of such eutectic fixed-points in a future ITS. Presently the accuracy of such measurements is partly limited by the large dimensions of the inevitable larger irradiance cells setting special demands to the applied high-temperature furnaces. However, this limitation will be solved by the new blackbody design presently developed by the VNIIOFI. 7. Application of the new high-temperature fixed-points in photometry, radiometry and thermometry 7.1. Photometry and radiometry 7.1.1. Standard sources for thermal radiation In 1948, with the first definition of the candela, Planckian radiators using the freezing and melting temperatures of a metal – platinum – were used to realise the luminous intensity. Due to improved mechanical and thermal stability the new hightemperature fixed-points can again be used to define the candela based on blackbody radiation at various temperatures. The advantage of the novel M(C)–C eutectics is the availability of significantly higher temperatures than the temperature of the platinum fixed-points of around 2042 K. The higher temperature results in significant higher spectral radiance resulting in lower calibration uncertainties. Additionally such fixed-points can be used to replace FEL lamps as transfer standards for irradiance and illuminance. It has been outlined in literature [113], that the spectral irradiance of some M(C)–C eutectics is at least of the same order or even larger than that of a FEL lamp. This is illustrated with a typical measurement geometry in Fig. 72, the resulted spectral irradiances at different distances are shown in Fig. 73. It has already been shown that M(C)–C cells with a diameter of the radiating cavity of 10 mm can be fabricated routinely and that such cells are sufficiently robust for inter-laboratory comparisons. The spectral irradiance of such cells applying TiC–C [114] and ZrC–C [115] has been measured by filter radiometers with central wavelengths around 650 nm by NMIJ, VNIIOFI and PTB for TiC–C and by PTB and VNIIOFI for the ZrC–C. The difference in the obtained melting temperatures at 3034 K (TiC–C) differs by only 0.24 K and at 3156 K (ZrC–C) by only 0.42 K. These differences in temperatures correspond to a difference in spectral radiance at 555 nm of 0.07% and 0.1%, respectively. These numbers indicate that the realisation of the candela by M(C)–C eutectics would already be more than a factor of two better than the realisation by the platinum blackbody according the CIPM SI definition of the candela from 1948.

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Fig. 72. Geometry of the application of M(C)–C eutectics as standards of irradiance and illuminance [113] (courtesy IOP Publishing Ltd.).

Fig. 73. Estimated spectral irradiance obtained in the A2 plane—comparison with lamps. (The distances for the FEL lamp and the deuterium lamp are 500 mm and 300 mm, respectively.) (a) AFP = 13 mm, A1 = 8 mm, A2 = 5 mm, d1 = 200 mm, d2 = 500 mm. (b) AFP = 9 mm, A1 = 5 mm, A2 = 5 mm, d1 = 200 mm, d2 = 500 mm [113] (courtesy IOP Publishing Ltd.).

However, M(C)–C eutectics can also easily replace the transfer or secondary standards for spectral radiance or luminance as well as for spectral irradiance or illuminance. Usually subjects for transferring these quantities are tungsten strip or filament lamps, typical examples are shown in Fig. 74. The radiometric quantities represented by the lamps shown in Fig. 74 are prone to large uncertainties, because the radiation emitted by the tungsten strip and the tungsten filament transmitted by the quartz window is highly polarized and dependent on the location on the tungsten strip, the angle of emission and on the wavelength as can be seen in Fig. 75. Additionally, the stability of such lamps is not sufficient and they have to be re-calibrated frequently [104]. In addition, when such lamps have to travel between several laboratories the instability and the risk of damage increases significantly. This fact is the critical point and the limiting factor for international comparisons for checking the agreement of the realisation of the radiometric and photometric units between national metrology institutes [117,

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Fig. 74. Transfer standards for spectral radiance and luminance (a) tungsten strip lamps of type Wi17 and spectral irradiance or illuminance (b) tungsten filament lamp of type Wi40 (manufactured by Osram [131]).

Fig. 75. Dependence of the spectral radiance realised by tungsten strip lamps. (a) dependence on location [51], (b) dependence on observation angle [104] and (c) dependence on band width and wavelength [116].

118]. In contrast, the M(C)–C eutectics have been shown to be highly stable, reproducible and robust [50]. Besides, as they are blackbody radiators, the emitted radiation is not polarised and the emission characteristic is that of an ideal Lambertian source and the emissivity is independent of the wavelength. What is necessary for proper operation is a furnace, capable to produce the temperature high enough to melt the respective material and having sufficiently good temperature homogeneity across the length of the cell. The development of such a practical furnace is the topic of an current project funded by the Federal Ministry of economics and labour of Germany [119]. Such a furnace in conjunction with a set of M(C)–C eutectic fixed-points can be used to transfer spectral radiance and irradiance, respective luminance and illuminance, over a wide range of magnitude and wavelength with a superior stability, reproducibility

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and accuracy. It has also been shown that the diameter of the radiating area of such M(C)–C eutectic fixed-point cells can be made as large 10 mm, significant beyond the homogeneous radiating areas of tungsten strip and filaments lamps. 7.1.2. Check and calibration of the spectral responsivity of photometers and radiometers Besides the realisation of the luminous intensity or other radiometric and photometric units, these fixed-points can also be used to check the calibration of photometers and radiometers by simply measuring the signal of the photometer or radiometer when viewing the blackbody during the melt of the respective eutectic. The signal of a filter radiometer unit according Eq. (1.9) ∞

Z

Lλ,s (λ, T ) s (λ) dλ

IPhoto = G · 0

with Lλ,s (λ, T ) defined by Eq. (1.3) Lλ,s =

c1

1

λ5 exp( λc2T ) − 1

can be approximated, in particular for a spectral responsivity of small bandwidth, by the following equation in its simplest form C

IPhoto =

2 exp( AT + )−1 B

c

(7.1)

with A, B, and C as adjustable parameters, describing the spectral characteristics of the spectral responsivity of the filter radiometer [120,121]. Eq. (7.1) is the result of a Taylor expansion of Planck’s law of radiation and application of Eq. (1.1). In the literature more general equations containing more components of the Taylor expansion have been presented [122]. According to the Taylor expansion, parameter C is connected to the spectral responsivity s(λ) according [122] ∞

Z C = c1 0

s (λ)

λ5

dλ.

(7.2)

Parameter A and B can be calculated according [122]

"

#
4µ4 − 9µ22 µ2 µ3 1 − 6 2 + 3 − 14 + ··· λ0 λ40 λ0

A = λ0

(7.3)

and B=

c2

"

2

#
4µ4 − 9µ22 µ2 µ3 −7 3 +7 + ··· λ20 λ40 λ0

(7.4)

with

R∞ λs (λ) dλ λ0 = R0 ∞ s (λ) dλ 0

(7.5)

and

R∞ µi =

0

(λ − λ0 )i s (λ) dλ R∞ . s (λ) dλ 0

(7.6)

Eqs. (7.1)–(7.5) can in principle be used to determine the spectral responsivity of the filter radiometers or the photometers, by simply measuring the signal of the filter radiometer or photometer when viewing under a defined geometry on at least three M(C)–C fixed-points. This is in particular possible for small bandwidth interference filter radiometers, as has already been shown in the 1970s for the application of such a method in the field of radiation thermometry [120]. A much more simpler application, also of great importance in photometry and radiometry, is the application of M(C)–C eutectics for the check of the stability of a photometer or a filter radiometer. For that purpose only one M(C)–C fixed-point is needed. Then the signal of this device according Eq. (1.9) is measured for one arbitrary M(C)–C eutectic – preferably at a temperature near the temperature of normal operation of the device – directly after the spectral responsivity calibration of the device. Then measurements of the same M(C)–C eutectic are performed in regular time intervals. As long as the obtained signal does not deviate from the originally measured value, within specified limits, no recalibration is necessary. Assuming a silicon photodiode-based photometer with a perfect V (λ) adopted spectral responsivity and assuming the fixed-point temperature of a TiC–C and a ZrC–C eutectic can be realised with an uncertainty as low as the differences measured in Refs. [114,115] the check of the calibration of such a photometer can be performed with an uncertainty of about 0.07% and 0.11%, without performing time consuming spectral responsivity measurements.

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7.2. Thermometry The original aim of the development of M(C)–C eutectics was to reduce the uncertainty of the International Temperature Scale of 1990 in the temperature range above 1357 K by making reproducible and stable temperature fixed-points available [29]. During the EU funded project HIMERT it has been successfully shown that M–C eutectics can be fabricated with sufficient robustness to perform frequent measurements over a long period of time. The results of an international comparison where five different M–C eutectics fabricated by three different institutions have been compared and the absolute thermodynamic temperatures of these cells have been determined showed very promising results (cf. Chapter VI). The repeatability of the cells was better than 100 mK, the reproducibility better than ±100 mK and the absolute temperatures have been obtained with uncertainties of about 200–300 mK, up to the Ru-C fixed-point, at a confidence interval of 95% (i.e. k = 2). The results of the absolute thermodynamic temperature have been compared between two national metrology institutes and agreed within the combined expanded uncertainty. Based on these auspicious results a major European project (Euromet/iMERA Project 926 [123]) is already started, with the final goal to assess the thermodynamic temperature of Co–C, Pt-C, and Re-C eutectic fixed-point as basis for a revision of the International Temperature Scale of 1990 [124]. This project is under auspices of the CCT WG5 Non-contact thermometry and is composed of the following work packages (WP) [the following part is taken from Ref. [124]]. WP 1: Cell reliability Establishing methods of construction for stable and robust cells. Undertaking ‘‘long-term’’ thermometry/radiometric reliability studies. Proposed time frame 2006–2008. WP 2: Cell reproducibility The primary cells (at least two per M(C)–C eutectic) used for thermodynamic temperature assignment (in WP 5), will be constructed by at least three different laboratories. Their melt temperatures will need to be measured to assign reproducibility values before WP 5 is initiated. This assignment does not need to be performed using absolute radiometry, only a relative measurement needed. These measurements should be performed in at least one laboratory capable of undertaking measurements to the required precision to determine the said reproducibility. The proposed time frame for this WP is 2007–2009, to run in parallel with WP 4, but to be finished prior to WP 5. WP 3: Methods for specifying the operational characteristics of M(C)–C eutectic fixed points This WP should run in parallel with the WP’s 1, 2 and 4, so the proposed time frame would be 2006–2009. Keeping track of the developments by the coordinator and exchange of information between the partners involved and coordinator would be the main features of this WP. An interim survey (mid 2007) and a final report (end 2009) are to be drafted by the coordinator. WP 4: Assessing absolute thermometry capability of participating laboratories Using currently available eutectics a comparison of absolute radiometry capabilities is to be undertaken, using at most three eutectic fixed-points, of which stability and robustness have been checked in WP 1. The three materials considered are Re-C, Pt-C and Co–C. Ti(C)–C was considered but rejected for this exercise as being to early in its development phase for inclusion. The proposed timeframe for this comparison is 2007–2008, with up to 6 laboratories participating. A preliminary comparison of this type has already been performed between PTB and NIST using NPL cells with promising results [50]. This type of comparison would clearly identify where limitations in current radiometry capability existed and where further work was required to improve facilities. In this work package, where validating the performance of institutes in measuring the thermodynamic temperature is the prime target, repeatability of the transfer cells is the main criterion. So only the requirements as regards cell reliability i.e. long-term stability and robustness need to be met. WP 5: Assigning thermodynamic temperatures within the context of a multi-lateral comparison After the lessons learnt from the comparison exercise in WP 4 have been implemented a second comparison should be performed to definitively assign thermodynamic temperatures to a set of selected M(C)–C eutectics, probably Re-C, Pt-C, and Co–C. With the completion of WP 2 definitive high quality reliable and reproducible metal-carbon eutectic fixedpoint cells should have been constructed by the temperature researchers. Provided the necessary level of consistency in results has been achieved these absolute radiometry measurements would produce the baseline thermodynamic temperatures for metal-carbon eutectic cells for years to come. In addition a defining fixed-point of the ITS-90 [Ag, Au or Cu] should be included in this comparison. The proposed time frame for this WP is 2009–2010, with again up to 6 laboratories participating. WP 6: Redefining temperature above the silver freezing point Analysis of the results obtained in WP5. Preparing a proposal to the CCT on redefining temperature above the silver point (through mise-en-pratique) to formally allow dissemination of T mediated through M–C eutectics (2011). Following this research program M(C)–C eutectics will be available and implemented in an improved International Temperature Scale around 2012. The implementation of the three high-temperature fixed-points, namely Re-C, Pt-C, and Co–C will reduce the uncertainty in realisation of temperatures up to 2800 K by a factor of about 5.

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8. Summary Starting with the problem of realising the photometric SI base unit candela and the uncertainty of temperature measurements above the temperature of freezing silver (1234.93 K) the fundamentals for the measurement of optical radiation have been described. Special emphasis has been given to the absolute calibration of the spectral responsivity of silicon photodiodes, the most widely used and the most accurate detector for optical radiation in the wavelength range from 300 nm up to 1100 nm. Models analytically describing the spectral responsivity and its temperature dependence have been developed and compared with experimental results. For absolute measurements of optical radiation the areas of the radiating surface and the area of the detecting surface have to be known exactly. Two methods for determining the area of optical apertures have been presented. The accuracy in the determination of the aperture areas depends on the diameter of the used aperture and is in the order of 2 parts in 105 for circular apertures with 20 mm in diameter and about 3 parts in 104 for apertures with 3 mm diameter. Applying a sophisticated technique for transferring the spectral radiance of the reference detector, a cryogenic radiometer, to a transfer trap detector and finally to the applied filter radiometers and photometers uncertainties as low as 2 parts in 104 have been obtained. Using these detectors novel high-temperature fixed-points have been thoroughly investigated and the thermodynamic temperatures of several M(C)–C eutectics have been measured for the first time with uncertainties below 0.5 K up to temperatures of about 3150 K. The presented work is vital for the assessment of the thermodynamic temperatures for novel high-temperature fixed-points to be implemented in a new International Temperature Scale, greatly decreasing the uncertainty for temperature measurements above 1400 K leading to a about fivefold lower uncertainty around 3000 K. Radiometry and, in particular, photometry will benefit from these improvements in high-temperature thermometry, as it will then be possible to realise the SI base unit candela by stable and accurate high-temperature fixed-points. The novel high-temperature fixed-point materials will lead to a significant advantage over the former realisation, which relied on the Pt fixed-point. Although it is not advisable to change the definition of the SI unit candela again, it might be highly appropriate to include one of the novel high-temperature fixed-points as a practical realisation in the instruction for the realisation of the candela, maybe forming part of a mise-en-pratique for the candela. Beside the realisation of the candela the new high-temperature fixed-point blackbodies are notably applicable as transfer standards for spectral radiance and irradiance, and the respective luminance and illuminance. This because they are much more stable and accurate than the commonly used tungsten strip and wire lamps. In particular, application of such fixedpoint radiators will substantially improve and reduce the cost of dissemination of the radiometric and photometric units to industry, because the time consuming and costly recalibration of the hitherto used tungsten strip and filament lamps will no longer be necessary. Acknowledgements First of all I would like to thank Prof. Kaase for his continuous support and for the fruitful discussions and amendments to this work. I would also like to thank Dr. Serick for his willingness to review the manuscript and for his support to bring me in close touch with photometry. Very warm regards go to the co-workers of the working group ‘‘High-temperature radiation thermometry’’ at the Physikalisch-Technische Bundesanstalt, especially to Klaus Anhalt, Stephan Schiller, Elzbieta Kosubek and Thomas Schönebeck. Stephan Schiller performs a considerable part of the measurements and Klaus Anhalt does an enormous amount of scientific investigations during his doctoral work in the field of the metal carbon eutectics. Elzbieta Kosubek performed numerous measurements and investigations on the area of the precision apertures used. Thomas Schönebeck had every time a technical solution to all mechanical problems. I would also like to thank my other colleagues at the Physikalisch-Technische Bundesanstalt, especially form the working group ‘‘Temperature radiation’’ and ‘‘Detector based radiometry’’, namely Dieter Taubert, Berndt Gutschwager, Beate Prußeit and Lutz Werner. Berndt Gutschwager, as an expert in radiation thermometry, gives lots of essential comments and advices to help me understand practical problems in radiation thermometry. Dieter Taubert was responsible for the spectral responsivity calibration of the filter radiometers at the spectral comparator facility and Lutz Werner was responsible for the calibration of the transfer detectors at the cryogenic radiometer RTCR. The continued support and the lively discussions with Jörg Hollandt, Head of the Department High-Temperature and Vacuum Physics of PTB and the continued support from Wolfgang Buck, Head of the Division Temperature and Synchrotron radiation of PTB was vital for the establishment of this work. Very special thanks to Rüdiger Friedrich who gave the work at the PTB laboratories a special spirit and had always a solution to all the tricky optical problems which had to be solved within this work. I would also like to thank H.J. Jung and Joachim Fischer, who brought me in contact with the fascinating but also cumbersome topic of radiation thermometry, radiometry, and photometry and who were my first teachers in that field. This work would not have been possible without the contribution from metrologists from all over the world, in particular Yoshiro Yamada from the National Metrology Institute of Japan, Dave Lowe from the National Physical Laboratory of the United Kingdom who supplied MC cells for the reproducibility and thermodynamic temperature measurements as well as valuable discussion. Mohamed Sadli from the LNE-INM/CNAM in France, who delivered cells, gave valuable discussion and helped for the measurement of the temperature distribution inside the HTBB. Mikhail Sakharov and Boris Khlevnoy from the All Russian Institute of Optics, supplying the irradiance M(C)–C eutectic fixed-point cells and gave valuable discussion. Very special thanks go to Graham Machin from NPL who plays a basic role in performing this work. On the one hand by

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coordinating the world wide investigation in the field of M(C)–C eutectics and on the other hand by reviewing the manuscript and giving very valuable comments. Last but nor least I would like to thank my family for their continuing support, their charitableness when I was only physically present but not mentally and their help in keeping me aware about the other side of life outside the laboratory and the office. Finally I would like to thank the Optical Society of America and the IOP Publishing Ltd. for their courtesy to use part of the following publications [50,61,63,106,113,125–127] for this review article. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57]

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