4 Design and calibration of the bolometers

4.1 Design

The bolometers have been developed at IAS, and benefitted from studies in the 40 mK - 150 mK range done for thermal detection of single events due to X-ray or $\beta$ sources (Zhou et al. 1993) or to recoil of dark matter particles (de Bellefon et al. 1996). The design (see Fig. 7) is that of a classical composite bolometer with a monolithic sensor as devised by Leblanc et al. (1978). The absorber is made of a diamond window (3.5 mm diameter and 40 microns thickness) with a bismuth resistive coating ( $R = 100\, \Omega$) to match optical vacuum impedance. The sensors were cut in a selected crystal of NTD Ge to obtain an impedance around $10\,{\rm M}\Omega$ at 150 mK (the effective temperature of the bolometers during these observations, because the thermal and atmospheric backgrounds load the bolometers above the 0.1 K cryostat temperature). The whole sensitive system is integrated in an integration sphere coupled to the light cone. Moreover, by using an inclined absorber with a larger diameter than the 2.5 mm diameter of the output of the light cone we finally increase the optical absorption efficiency $\eta$ from 40 to 80%, before residual rays go out of the integration sphere (see Eq. 5). Indeed, the optical efficiency can be estimated with the well-known Gouffe's formula (Gouffe 1945), by considering that the effective cavity surface S is twice the surface of the resistive bismuth coating (which has an emissivity larger than 0.4: Carli et al. 1981). The entrance surface s is the 2.5 mm diameter output of the cone. With S/s= 2 (3.5/2.5)²=3.9, the final cavity emissivity is larger than 0.8. An additional internal calibration device (Figs. 7 and 3: a near infrared light fed by a diode on the back of the bolometer via an optical fibre) was sucessfully tested but not subsequently used, once the optics was available.

Figure 6: Schematic drawing of the Diabolo open-cycle dilution refrigerator. See Sect. 3. The cold plate is at 100 mK. The lHe is pumped in order to reach a temperature of 1.8 K which is required by the dilution fridge

Figure 7: Schematic drawing of one bolomotric detector with its optical Winston cone (1) and its integration sphere (2) (4 mm diameter). The diamond substrate (3) is coated on the rear surface with a resistive bismuth layer (square impedance of $100\,\Omega$ to maximise absorption). It is tilted in order to prevent that the first reflexion goes directly out. An optical fiber (4) (made of a bundle of 20  $\,\mu {\rm m}\,$silica fibers) of 180  $\,\mu {\rm m}\,$external diameter enters the sphere, permitting control of the response stability at any time. A low pass thermal filter (5) made of a low thermal conductivity stainless tube (5 mm diameter; wall thickness $250~\hbox{$\,\mu {\rm m}\,$ }\!\!$) with a cut-off at about 0.2 Hz supresses the noise coming from temperature fluctuations of the dilution cryostat. A 50  $\,\mu {\rm m}\,$gap in (6) is necessary to prevent short circuit of the thermal filter

4.2 Calibration

The theory of responsivity and noise from a bolometer has been written by Mather (1984) and Coron (1976). At the equilibrium, the Joule power dissipated in the bolometer, $P_{\rm J}$, and the absorbed radiation power, $P_{\rm R}$, are balanced by the cooling power $P_{\rm c}$ due to the small thermal link to the base temperature:

$$P_{\rm c}(T_1, T_0)= P_{\rm J}+ P_{\rm R},$$ (1)

with

$$P_{\rm c}= \frac{A}{L} \int^{T_1}_{T_0} \kappa(T) {\rm d}T= ... ...ght) ^\alpha - \left( {T_0\over T_g} \right) ^\alpha \right),$$ (2)

where A, L, and $\kappa$ are respectively the cross section, length and thermal conductivity of the material which makes the thermal link. We have approximated $\kappa(T)$ with a power law, $\kappa(T) \propto T^{\alpha-1}$. From the I-V curves, we find that for the reference temperature of $T_g= 0.1 \rm\,K$ the value of g and $\alpha$ are typically of g=140 picoWatts and $\alpha=4.5$ for both bolometers. The impedance can be approximated with:

$$R(T)=R_\infty \exp((T_{\rm r}/T)^\beta),$$ (3)

where $T_{\rm r}$, $R_\infty$ and $\beta$ are respectively 200 K, 0.80 $\Omega$, and 0.38 for channel 1 and 20 K, 52 $\Omega$, and 0.51 for channel 2. The electrical responsivity at zero frequency was deduced from the I-V curves using

$$S_{\rm el}(0) = \frac{Z-R}{2RI},$$ (4)

where $Z = {\rm d}V/{\rm d}I$ is the dynamic impedance calculated at the bias point on the I-V curve. We find electrical responsivities of the order of $3 \ 10^7$ and $20 \ 10^7 \rm\, V/W$ respectively under the sky background conditions (a load of one to few hundred picoWatts). With a noise equivalent voltage of typically $30 \rm\,nV \, Hz^{-1/2}$ above 2 Hz, the electrical NEP is approximately of $10 \ 10^{-16}$ and $2 \ 10^{-16} \rm\,W Hz^{-1/2}$ for channel 1 and 2 respectively. The response of the bolometer to the optical signal is linked to the electrical response via

$$S_{\rm opt}= \eta S_{\rm el},$$ (5)

where $\eta$ is the optical efficiency. Thus, assuming $\eta \ge 0.8$, the optical NEP (although not measured) at zero frequency could be reliably estimated to be better than $15 \ 10^{-16}$ and $3 \ 10^{-16} \rm\,W Hz^{-1/2}$ for channel 1 and 2 respectively.

The bolometer also responds to the base plate temperature fluctuations with a responsivity that can be deduced from the previous formalism:

$${{\rm d}V\over {\rm d}T_0}= -S_{\rm el} {{\rm d}P_{\rm c}\ove... ...{\rm el} g \alpha}{T_0} \left( {T_0\over T_g} \right) ^\alpha.$$ (6)

The 2 bolometers that we use have typical sensitivities to the base plate temperature of 0.2 and 1.2 $\mu {\rm V}/\mu {\rm K}$ respectively. We see from Eq. (6) that the larger the conduction to the base plate and the larger the sensitivity of the bolometer to an external signal, the most sensitive will the bolometer be to the fluctuations of the base plate temperature. As the base plate temperature T₀ fluctuates by typically $10 \rm\,\mu K$ over time scales of few seconds, a regulation of this temperature should be made in the near future to minimise fluctuations.

The time constant is less than 10 milliseconds for both bolometers as measured with particles absorbed by the bolometers against a small radioactive source.

   
4.3 The need for 0.1 K temperature in ground-based experiments

The background is relatively large in the case of ground-based experiments. There is a general prejudice that very low temperatures are thus not needed. Actually, the temperature required for optimised bolometers depends only on the wavelength, because the photon and bolometer noises both increase as the square root of the incoming background. The general formula is (Mather 1984; Griffin 1995; Benoit 1996):

$$T_{\rm max}= {{hc}\over{k}} {p\over\lambda},$$ (7)

where $hc/k= 14.4 \rm\,K$ mm and p is a dimensionless constant. It turns out that for classical bolometers with a resistive thermometer, one has typically $p\simeq 0.025$, so that the maximum temperature for millimetre continuum astronomy is 0.4 K. Allowing for non ideal effects and bolometers which would be 0.7 less noisy than the background noise, a temperature of 0.1 K is required in the 2 mm cosmological atmospheric window. The ultimate noise equivalent power for a given background P and temperature T is then

$${\rm NEP}/({\rm W Hz^{-1/2}})\simeq 10^{-17}(P T/ (10^{-13} {\rm\,W K}))^{0.5}.$$ (8)

The present bolometers are within a factor 3 of this limit, which is also the photon noise limit, thus leaving some margin for improvements.

Figure 8: Principle of Diabolo readout electronics. The square wave bias is adjusted to oppose the bias coming from the capacitive integrated current. Once these parameters are fixed, the out-of-equilibrium signal at the middle of the electrical bridge follows the varying radiation power absorbed by the bolometer

Figure 9: Mars observed with Diabolo at 1.2 mm at MITO. Contours are at 20, 40, 60, 80, and 100 mK$_{\rm RJ}$ brightness levels

Figure 10: Mars observed with Diabolo at 2.1 mm at MITO. Contours are at 20, 40, 60, 80, and 100 mK$_{\rm RJ}$ brightness levels

Figure 11: Orion observed with Diabolo at 1.2 mm at MITO. Contours are at 10, 20, 30, 40 mK$_{\rm RJ}$ brightness levels

Figure 12: Orion observed with Diabolo at 2.1 mm at MITO. Contours are at 10, 20, 30, 40 mK$_{\rm RJ}$ brightness levels