Absolute Calibration in FIR Radiometric Measurements

M. Zerbini and G. Galatola-Teka ENEA, Fusion and Nuclear Safety Department, C. R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy

variables is made by imposing the conservation of energy, for I. INTRODUCTION example: truly rigorous and quantitative approach to Black Body radiation is the fundamental starting point of !" ! A �(�)�� = �(�)�� so: �(�) = �(�) = �(�) experimental Radiometry in the region of Far Infrared and !" !! Millimeter waves (FIR). Absolute (i.e. calibrated) measurements of electromagnetic radiation are widely used in The definition of the blackbody spectral quantities, according many areas of Physics, from Plasma Diagnostics in Nuclear to Boyd’s [1], is given below and summarized in Table 1.

Fusion, to Radio Astronomy and Atmospheric Physics. Along - Spectral Radiant energy density u: radiant energy per unit the years different Authors have used personalized constants frequency contained in a volume dV and definitions, sometimes arbitrary, and the choices were - Spectral Radiant Exitance M: flux per unit area and often hidden into papers and analysis software, hence making frequency, leaving the surface of a source impossible to compare properly the results, even within the - Spectral Irradiance E: flux per unit area and frequency, same field of research. In Boyd’s words [1]: received by a surface element (E=-M) - Spectral Radiance L: flux per unit projected area per unit "Considerable care should be exercised when comparing the results of various Authors, since the same word sometimes is used to denote different solid angle and unit frequency leaving a source physical properties" - Spectral Radiant intensity I: flux per unit solid angle and frequency emitted by a source in a given direction The aim of this work is to start from original Planck's theory [2], supply a set of rigorous definitions and units of the Any given integral quantity A is related to the corresponding quantities involved in Radiometry absolute calibration based spectral one Aν by: on reference [1], then on this basis compare the equations ! from various Authors. A special discussion is reserved to the � = �!�� blackbody applications for Electron Cyclotron Emission ! (ECE) diagnostics. Finally, important practical applications, -1 mainly in the area of Plasma Diagnostics, will be discussed hence Aν contains a Hz factor in the dimensions. with the support of experimental data. The spectral range discussed in this work extends from about 50 GHz till Quantity Symbol Definition Units Spectral Radiant energy u ��/�� J m-3 Hz-1 1000 GHz (~5mm down to ~300 µm wavelength). density Spectral Radiant Exitance M �Φ/�� W m-2 Hz-1 Spectral Irradiance E �Φ/�� W m-2 Hz-1 ! -2 -1 -1 II. BLACKBODY RADIATION BASIS Spectral Radiance L � Φ/��!"#$�Ω W m sr Hz -1 -1 A Black Body in thermal equilibrium is described by its Spectral Radiant intensity I �Φ/�Ω W sr Hz Spectral Energy density (i.e. per unit frequency), measured in Table 1: Spectral Radiometric quantities and corresponding SI units. -3 -1 [J m Hz ] in the International System of Units (SI). In 1914 Planck demonstrated {[2] p. 168, eq. 275}, that for a Boyd's black-body unpolarized Spectral radiant energy blackbody in thermal equilibrium with the surrounding density identically coincides with Planck's internal energy radiation the internal spectral energy density for density eq. (1a) without adjustment {[1] p. 48 eq. 3.65 with monochromatic unpolarized radiation of frequency ν is: refractive index N=1 to simplify notation}. With basic arguments about isotropic flux of electromagnetic energy ! !!!! ! -3 -1 flowing at light velocity c Boyd shows that {[1] p. 30,31} �(�) = [J m Hz ] (1a) !! !!!! !"!! since the energy passing through an element dA in the time interval dt fills a cylinder of base dA and height cdT, we have: ℏ!! ! �(�) = [J m-3 rad-1 s] (1b) ! !!!! !!!! !!"#!! � = � [W m-2 sr-1 Hz-1] (2) !! c is the light velocity, h and k are Planck and Boltzmann and integrating over the inward hemisphere: constants and ℏ=h/2�. Eq. (1b) is expressed in terms of ! radian-frequency ω=2��. The conversion between spectral � = �� hence � = � [W m-2 Hz-1] (3) ! With the above relationship and using u from (1a): which is fully coherent with Costley “ECE Intensity” in [10]. The absolute value of thermal emission ECE is measured by !" !"!! ! calibrated Martin-Puplett Michelson Spectrometers [12]. �(�) = = [W m-2 sr-1 Hz-1] (4) !! !! !!! !"!! Calibration is performed by using 2 blackbody sources to remove the background signal by difference. Specac blackbody with T=843 K [13] and Liquid Nitrogen (LN2) at !" !!!!! ! -2 -1 T=77 K give a differential calibration temperature T=766 K. �(�) = = ! !! !" [W m Hz ] (5) ! ! ! !! RJ approximation applies to both calibration sources, although

only marginally for LN2. Since large area blackbodies fills the III. COMPARISON WITH OTHER AUTHORS ECE antennae we refer to the hemispherical irradiance E=�L. Let’s compare the blackbody radiation expressions from The Antenna active area (16 cm2 in this case) is required to different Authors. Note that Planck and Boyd deal with calculate the absolute power received, but since this parameter unpolarized radiation. If single polarization is used their remains obviously the same both in calibration and plasma expression have to be divided by 2. Lengyel [3] introduces the measurements, it can be included in the calibration function "Radiation escaping the cavity through a hole cut into the F=Theoretical Spectrum/Calibration Spectrum. An example walls per unit area of the hole". This radiative flux density at of ECE calibration spectra for FTU Tokamak is shown and the exit of the cavity is appropriately called "black-body commented in Fig. 1. Given the very low absolute value of radiation” and corresponds identically to Boyd’s irradiance of FIR power (pW in calibration and nW with plasma) optimised high-directivity and high-gain antennas are necessary to attain the cavity wall E in eq. (3), converted in wavelength λ. practically usable values of SNR [14]. Kimmit in his historical FIR book {[4] eq. (3.1) p. 43} introduces the "flux of radiation per unit source area per hemisphere per unit wavelength interval". Kimmit assumes unpolarized radiation (2-polarizations). The above definition and expression are coherent and identical to the Boyd's hemispherical irradiance E(λ) converted in terms of wavelength. Coherence of blackbody expressions between other Authors [5,6,7,8,9], can be similarly demonstrated.

IV. ECE RADIOMETRY The Tokamak magnetic field B varies along the major radius ! R with the relationship: � = ! � , where the subscript 0 ! ! indicates the torus axis. Plasma electrons in each radial !" location emit ECE at frequency � = ∼ 28���/�����. Since !! ! Fig. 1. Calibration source theoretical (a) and measured (b) spectra. The shape in the relevant spectral region (50 GHz to 1000 GHz) the of the measured spectrum (b) is due to transmission features of the 2nd harmonic follows blackbody laws, the absolute value of spectrometer optics and to the InSb detector responsivity, remarkably linear local Maxwellian Temperature, i.e. the electron temperature over this large temperature range [12]. Water vapour absorption lines are also present. The calibration function F is the ratio between the two spectra. (c): profile, can be obtained from a direct measurement of ECE ECE spectrum of FTU pulse #35496 with B0=6T at 0.5 s, calibrated using F. nd emission. This possibility made ECE a fundamental diagnostic In the shaded 2 harmonic optically thick region (fce0=340 GHz) the in nuclear fusion research [10]. Heald & Wharton [11] hemispheric irradiance is consistent with blackbody emission at 20 MK introduce for ECE the “unpolarized radiation density in (~2 KeV) which is the average plasma temperature in this pulse. thermodynamic equilibrium with matter at temperature T”, in the radian-frequency interval dω: REFERENCES [1]. RW Boyd, “Radiometry & Detection of Optical Radiation”, Wiley (1983). ℏ!! ! [2]. M Planck, M Morton, “The Theory of Heat Radiation”, Blakiston (1914). � �� = �� (6) ! !!!! !ℏ! !"!! [3]. B.A. Leyngel, “Introduction To Laser Physics”, Wiley (1966). [4]. M.F. Kimmitt, 'Far-infrared Techniques', Pion (1970). [5]. E. Bründermann et al., “Terahertz Techniques”, Springer (2012). Uω is identical to Plank (1b) for u(ω) in radian-frequency. In [6]. S. Erkoç “Fundamentals of Quantum Mechanics”, Taylor-Francis (2007). current fusion experiments plasma temperature is in the region [7]. E. Hecht, A. Zajac, “Optics”, Addison Wesley (1974). 1-10 KeV (1KeV ~107 K) so ℎ� ≪ �� or � ≪ 21 ���/� so [8]. BM Javorskij, AA Detlaf , “Handbook of Physics ” (Справочник по физике), MIR (1972). that Rayleigh-Jeans (RJ) approximation can be applied: [9]. B.E.A. Saleh, MC Teich, “Fundamental of Photonics”, Wiley, (1991). making use of eq. (2) and dividing by 2 for the 1-polarization [10]. A. Costley, in “Diagnostics for Fusion Reactor Conditions”, Varenna, case, eq. (6) converted in RJ approximation becomes: EUR 8351-I (1982). [11]. M.A. Heald, C.B. Wharton, “Plasma Diagnostics with Microwaves”, Wiley (1965). !!!" �(�) = (7) [12]. National Physical Laboratory, “Radiometric Calibration of SPECAC !!!!! 40.110 MM wavelength large area source”, January, 1990. [13]. P. Buratti & M. Zerbini, Rev.Sci.Instrum 66, 4208 (1995). [14]. C.A. Balanis, ‘Antenna Theory: analysis and design’, Wiley, (2005).