3.9 Noise
3.9.1 Noise Model
We can think of noise as a random variable added to our true signal. Here we are using the variablei to indicate that this is a current noise:
imeas(t) =i(t) +ni(t) (3.4)
hnii= 0,hn2ii=σn2i, (3.5) where σni is the variance. The following equation describes the power spectral density (PSD) of the noise:
σn2i = Z ∞
0
i2n(f)df. (3.6)
The variance for Gaussian noise is uncorrelated as a function of frequency. Variances from uncorrelated noise sources add in quadrature. So for each component of the noise model, we will estimate a value and add the contributions in quadrature.
In order to understand the SPIDER noise, a noise model was developed to describe each component of the noise. The noise components include photon, phonon (thermal fluctuation), Johnson, excess and amplifier noise. This section is based on work done by Jeff Filippini. The noise model for TES bolometers is described in full detail in [51].
3.9.1.1 Photon Noise
Shot noise in electronic systems originates from the fact that current actually consists of a flow of independent quanta (electrons). Shot noise is typically both temperature- and frequency-independent. Photon noise is the same phenomenon as shot noise, only it originates from the (quantized) particle nature of light. We expect photon noise to be the largest contribution to the unaliased noise in the science band of the SPIDER detectors.
Photons with Bose correlations in arrival times will add to the usual shot noise. The noise equivalent power (NEP) due to photon noise is
NEP2photon= 2hhνiPopt+Popt2
2∆ν, (3.7)
wherePoptis the optical power on the detectors,hνiis the band center, ∆νis the bandwidth, and h is the Planck constant. Here the first term is considered the “shot” noise term and the second term is the “Bose” noise term. If the system is under flight-like loading, we are typically dominated by the shot noise term. If we are open to the room and on the aluminum transition, it turns out that the Bose noise term dominates.
For flight-like loading we can estimate a value for the contribution to the overall noise from photon noise by using the following typical values: Popt = 0.9 pW, ν = 150GHz, and
∆ν= 37.5 GHz. Then,
NEPphoton≈14 aW/
√
Hz. (3.8)
3.9.1.2 Johnson Noise
Johnson noise (sometimes called Johnson-Nyquist noise) is a voltage noise from the Brow- nian motion of charge carriers inside an electrical conductor at equilibrium (i.e., regardless of applied voltage). It is a specific result of the Fluctation-Dissipation Theorem, which states that any dissipative element experiences temperature-dependent fluctuations due to coupling to thermal degrees of freedom.
The voltage noise due to Johnson noise is
vn2 =h(V − hVi)2i= 4kT R, (3.9) wherekis Boltzmann’s constant,T is the resistor’s temperature, andR is the resistance of the resistive element.
There are two sources of Johnson noise in the TES circuit: the shunt resistor (Rsh) and the TES resistance (RT ES) [51, 20]:
NEP2Johnson = NEP2sh+ NEP2TES (3.10)
= 4kTshRshITES2 (L1−1)2
L21 + 4kT RTESITES2 1 L21,
whereL1is the loop gain. Note that the Johnson noise of the TES resistor goes as the inverse of the square of the loop gain, which makes the DC contribution from this component several orders of magnitude smaller than that from the shunt resistor. This is due to the fact that the Johnson noise on the TES resistor is coupled to both the electrical and thermal circuits, and their correlated contributions cancel below the electrothermal feedback frequency.
We use the following typical values for the parameters in Eqn. 3.10 to get an estimate of the NEP contribution from Johnson noise: Rsh = 0.3mΩ, RTES ≈ 20mΩ, Ish ≈ 8µA, Tsh= 350mK, andL1 ≈20.
We find
N EPJohnson≈2aW/√
Hz. (3.11)
3.9.1.3 Thermal Fluctuation Noise
Thermal fluctuation noise is also known as phonon noise. Phonon noise arises from the random exchange of phonons between a thermal mass and its environment.
In analogy with Johnson noise, a thermal conductance has an equivalent power noise:
NEP2TFN= 4kT2Gγ(T, Tb). (3.12) where γ is a correction factor, k is the Boltzmann constant,G is the thermal conductance of the link,T is the temperature of the detector, andTb is the temperature of the bath. The correction factor γ depends on the exponent of thermal conductance and on whether the phonon reflection from boundaries is specular or diffuse. It is a function of the temperature of the TES and typically takes a value between 0.5 and 1. It can be thought of as accounting for non-equilibrium temperature differences across the thermal link.
A rough number for this contribution to the noise for a SPIDERdetector can be found using the following values: G = 15pW/K, γ ≈0.5, and T≈500mK.
So,
NEPTFN≈10 aW/
√
Hz. (3.13)
Excess noise at frequencies above our science band (see §3.9.1.5) indicate that that the above equation is describing an oversimplified model of the SPIDER TESs. The model should include internal thermal fluctuations across internal island conductances that look like excess Johnson noise. Additional complications to the model may include supercon-
ducting percolation effects or some other finite, temperature-dependent coupling.
3.9.1.4 Amplifier Noise
Amplifier noise is a significant source of noise in the SPIDERsystem, especially while dark.
It is approximately white to 2MHz and has a 1/f component. Both the warm and cold electronics contribute to amplifier noise. Much of the amplifier noise in the SPIDERsystem is aliased as a result of an avoidable property of the time-domain multiplexing used to read out the detectors. Although the detectors are only read out at 15kHz, the system must have the bandwidth to switch between detector rows every∼2µs and wait for the SQUIDs to settle before taking data. The upper limit of the bandwidth of the system is set by the readout card of the MCE, which has has analog-to-digital converters (ADCs) that operate at 50MHz.
If we sample at frequency fs, all noise at f > fs/2 is aliased back to f → |f −fs/2|
and the total noise variance is unchanged. The increase in current noise due to aliasing is proportional to the square root of the ratio of the bandwidth (BW) to the sampling frequency:
in(f)→in(f) s
BW
fs/2. (3.14)
For SPIDER,fs = 15.15 kHz and thus aliasing boosts SQUID noise by approximately 12 times. While this is not the leading contribution to the overall noise of the SPIDER detectors, it leads to a total NEI for the amplifier noise of approximately 45 pA/√
Hz. We can multiply this by a typical value for the responsivity of a detector dP/dI = 0.13 µV to get
N EPamp ≈4aW/
√
Hz. (3.15)
A fair amount of work was done by NIST (who create the SQUIDs used in the SPIDER system) to decrease aliasing by changing the SQUID design. This included reducing the SQUID noise current by adding input coil turns, reducing the SQUID bandwidth, and reducing the detector normal resistance. These changes improved the uniformity of the noise of on the detectors at science frequencies, although they did not improve the total noise of the lowest noise detectors.
Table 3.3: Noise Budget for 90GHz and 150GHz detectors Photon Johnson Phonon Amplifier Total
90GHz 11 2 10 4 16
150GHz 14 2 10 4 18
All values have units of aW/√ Hz.
3.9.1.5 Excess Noise
Our measured noise somewhat exceeds the noise predicted by the model, especially at kilohertz frequencies where we see an “excess noise bump” in our noise spectra (see the grey spectrum in Fig. 3.12). Excess noise is a known feature of TES systems and is not fully understood. There are several plausible explanations for it: internal thermal fluctuations, phase transition effects, and nonlinear or non-equilibrium behavior. The main concern of excess noise is that it will be aliased back into the science band, since a significant amount of power from the excess noise extends beyond the Nyquist frequency of our readout system.
As described in the previous section, several changes were made to the readout system to reduce the aliasing from this noise.