50 100 150 0
0.5 1
ν [GHz]
F(ν)
0 1 2 3
0 0.5 1 1.5 2
∫ (F(ν)hν(ehν/kT−1)−1dν) [pW]
P Joule [pW]
a = 0.45
50 100 150
0 0.5 1
ν [GHz]
η(ν)
Figure A.1: (Left)F(ν) is measured by FTS, and arbitrarily normalized. (Center) A simple linear fit to the data taken at different load temperatures determines the normalization pref- actor “a” according to Equation A.4. (Right) The calculated prefactor is used to normalize F(ν) to findη(ν).
the remainder of this section.
A complete measurement of optical efficiency requires passband measurements to define the spectral shape ofη(ν). Fourier transform spectroscopy (§4.2) provides an unnormalized passbandF(ν). Using Equations A.2 and A.3, we find
η(ν) =aF(ν) =− dPJ oule
d
F(ν)hν(ekThν −1)−1dν
F(ν). (A.4)
The prefactorais calculated by a simple linear fit to the data. As an example, in Figure A.1 I examine my measurements of the device designated by JAB080529 (3, 2, B).
In the event that spectroscopy is not available, I isolate the magnitude of the integrated optical response from the details of the spectral response by defining
η0≡
η(ν)dν 0.25ν0
, (A.5)
whereν0is the target band center. I choose this definition because the target bandwidth is 25%. Then
Qη0
1.125ν0 0.875ν0
hν0
ehν0/kT −1dν, and (A.6)
η0=− dPJ oule
d1.125ν0
0.875ν0 hν(ekThν −1)−1dν
. (A.7)
This is a good approximation assuming hν/(ehν0/kT −1) does not vary significantly over
the passband, which is true unless there is significant response out of the primary band.
Sometimes we wish to express optical response with units of power over temperature.
However, the absorbed power is not linearly related to the physical temperature. To work in these units we therefore instead convert the incident power in to a Rayleigh-Jeans “tem- perature,” given by
TRJ ≡k−1(0.25ν0)−1
1.125ν0
0.875ν0
hν0
ehν0/kT −1dν
. (A.8)
This is not an actual physical temperature. It corresponds to the temperature of the ide- alized Rayleigh-Jeans source (hν kT) that would radiate the same amount of power in a 25% wide band around ν0 as our actual blackbody source. Just as was done above,
|dPJ oule/dTRJ| is calculated using a simple linear fit to the data (see, for example, Fig- ure 4.17).
We often quote|dPJ oule/dTRJ|in tabulations of measured optical response. These values can be expressed in terms ofη using
η0= (0.25ν0k)−1 dPJ oule
dTRJ
, and (A.9)
η(ν)dν=k−1 dPJ oule
dTRJ
. (A.10)
A.2 Alternative Methods for Calculating Internal Loading
The More Robust Method
This technique uses all of the same data sets associated with the basic method and requires two additional data sets. Each additional data set consists of load curves taken at a variety of focal plane temperatures (∼10 different temperatures between the fridge base temperature and the device saturation temperature). One set is taken on a dark run (Q0), and the other is taken on a cold load run at a constant (preferably low) cold load temperature.
Each of these two data sets can be used to fit the parameters of thermal conductance (see Figure 3.12 for a reminder). The results should agree. For the cold load data, the constant
300 350 400 450 500 550
−2
−1 0 1 2 3 4
Tsub [ mK ] P Joule in transition [ pW ]
Tc
−Q Dark data
Cold load data
Figure A.2: The optical loadQabsorbed by devices in the cold load configuration introduces a simple DC offset inPJ oule (independent ofTsub). The curve for the dark data intercepts the temperature axis atTsub =Tc, so the projectedPJ oule for the fit to the cold load data is−QatTsub=Tc.
nonzero optical power introduces a simple DC offset in power (−Q).
PJ oule = Plegs−Q (A.11)
= GT0
β+ 1 Tc
T0
β+1
− Tsub
T0
β+1
−Q
We can extractQfrom our thermal conductance fit for the cold load data (Figure A.2).
Q=−PJ oule for fit projection at Tsub =Tc (A.12) A little algebra produces another equivalent expression forQ,
Q= GT0
β+ 1 Tc
T0 β+1
− Ti
T0 β+1
, (A.13)
whereTi is the temperature axis intercept for the curve fit to the cold load data.
The optical power Q is the sum of the loading from the cold load and the internal loading. We subtract the cold load contribution to calculate the internal loading,
Qint=Q− dP
dTRJ
TColdLoad,RJ. (A.14)
This alternative method reduces statistical errors by using∼10×as much load curve data as the simple method. In addition, it greatly suppresses systematic errors due to small inaccuracies in electrical calibration factors (e.g., wiring resistances, SQUID gain) that may bias PJ oule in different ways from one run to the next. For example, a calibration uncertainty resulting in a 5% disagreement in powers between dark data and cold load data would result in an internal loading systematic error of 0.05*PJ oule(or about 0.15 pW) using the simple method. Using this alternative method the systematic error would simply scale the loading result, so the effect would be no more than 0.02 pW for optically coupled pixels and 0.002 pW for dark TES.
Without Dark Data
Generally, without dark data there is no way to break the degeneracy between internal loading andTc. However, there are four dark TES (not coupled to antennas) on each tile.
Dark TES exhibit very low internal loading (see Appendix B.10). Neglecting this small loading, we can calculateTc for the dark TES using only cold load data. For dark TES, the correction due to optical response to the cold load is small (∼1 mK, see Appendix B.8), but we make it anyway.
Tc=
Ti,darkβ+1 +T0β(β+ 1) G
dP dTRJ
dark
TColdLoad,RJ
1/(β+1)
(A.15)
The dark TES bolometers are fabricated in the same way as the optically coupled devices, so they should exhibit similar values ofTc. We can therefore, with some caution, take the average measuredTc for the dark devices and substitute it into Equation A.13 to calculate the loadingQfor the optically active devices.