2
Some Basics
2.1 Measurement
Measurement is a physical process to determine the value (magnitude) of a quantity. The quantity value can be expressed as
2.1
where {q} is the numerical value and [Q] the unit (see following chapter). Repeated measurements of the same quantity, however, generally will result in slightly different results. In addition, systematic effects affecting the measurement result might be present and have to be considered. Thus, any measurement result must be completed by an uncertainty statement. This measurement uncertainty quantifies the dispersion of the quantity values being attributed to a measurand, based on the information used. Measurement uncertainty comprises, in general, many components. Some of these may be evaluated by type A evaluation of measurement uncertainty from the statistical distribution of the quantity values from series of measurements and can be characterized by standard deviations. The other components, which may be evaluated by type B evaluation of measurement uncertainty, can also be characterized by standard deviations, evaluated from probability density functions based on experience or other information. For the evaluation of uncertainties of measurements, an international agreed guide has been published jointly by ISO and the Bureau International des Poids et Mesures (BIPM), the Guide to the Expression of Uncertainty in Measurement (GUM) [1, 2]. Precision measurements generally are those with the smallest measurement uncertainty.
2.1.1 Limitations of Measurement Uncertainty
One might tend to believe that measurement uncertainty can be continuously decreased as more and more effort is put in the respective experiment. However, this is not the case since there are fundamental as well as practical limitations for measurement precision. The fundamental limit is a consequence of the Heisenberg uncertainty principle of quantum mechanics, and the major practical limit is due to noise.
2.1.1.1 The Fundamental Quantum Limit
Note that throughout this book, we will use the letter f to denote technical frequencies, while the Greek letter ν is used to denote optical frequencies.
The Heisenberg uncertainty principle is a fundamental consequence of quantum mechanics stating that there is a minimum value for the physical quantity action, H:
where h is the Planck constant. Action has the dimensions of energy multiplied by time and its unit is joule seconds. From the Heisenberg uncertainty principle, it follows that conjugated variables, like position and momentum or time and energy, cannot be measured with ultimate precision at a time. For example, if Δx and Δp are the standard deviation for position, x, and momentum, p, respectively, the inequality relation
2.3
holds (). Applied to measurement, the argument is as follows: in the course of a measurement, an exchange of information takes place between the measurement system and the system under consideration. Related to this is an energy exchange. For a given measurement time, τ, or bandwidth of the measurement system, , the energy which can be extracted from the system is limited according to Eq. (2.2) [3]:
2.4
Let us now consider, for example, the relation between inductance, L, and, respectively, magnetic flux, Φ, and current, I (see Figure 2.1). The energy is given by , and consequently,
2.5
Figure 2.1 Components and quantities considered (left) and the minimum current, Imin, and the minimum magnetic flux, Φmin, versus inductance, L, for an ideal coil.
(From [3], with kind permission from Wiley-VCH.)
These relations are depicted also in Figure 2.1. The gray area corresponds to the regime which is accessible by measurement. Please note that this is a heuristic approach which does not consider a specific experiment. Nevertheless, it may provide useful conclusions on how to optimize an experiment. For instance, if an ideal coil (without losses) shall be applied to measure a small current, the inductivity should be large (e.g., , , and then ). If instead the coil is applied to measure magnetic flux, L should be small (e.g., , , and then where is the ).
Likewise, for a capacitor with capacitance, C, the energy is given by
2.6
and thus,
2.7
Finally, for a resistor with resistance, R, the energy is given by
2.8
and thus, for the minimum current and voltage, respectively, we obtain
2.9
2.1.1.2 Noise
In this chapter, we briefly summarize some aspects of noise theory. For a more detailed treatment of this important and fundamental topic, the reader is referred to, for example, [4].
Noise limits the measurement precision in most practical cases. The noise power spectral density, P(T, f)/Δf, can be approximated by (Planck formula)
2.10
where f is the frequency, kB the Boltzmann constant, and T the temperature. Two limiting cases can be considered as follows.
Thermal Noise (Johnson Noise) (kBT ≫ hf)
2.11
According to this “Nyquist relation,” the thermal noise power spectral density is independent of frequency (white noise) and increases linearly with temperature. Thermal noise was first studied by Johnson [5]. It reflects the thermal agitation of, for example, carriers (electrons) in a resistor.
Quantum Noise (hf ≫ kBT)
2.12
The quantum noise power spectral density in this limit is determined by the zero point energy, hf, and is independent of temperature and increases linearly with frequency.
Thermal noise dominates at high temperatures and low frequencies (see Figure 2.2). The transition frequency, fc(T), where both contributions are equal depends on temperature and is given by
2.13
Figure 2.2 Noise power spectral density, P(T, f), versus frequency for different temperatures.
(From [3], with kind permission from Wiley-VCH.)
This transition frequency amounts to 4.3 THz at and 60.6 GHz at the temperature of liquid He at .
The thermal noise in an electrical resistor at temperature T generates under open circuit or shortcut, respectively, a voltage or current with effective values:
2.14
2.15
To keep the noise level low, the detector equipment should be cooled to low temperatures to reduce thermal noise. Going from room temperature (300 K) to liquid He temperature (4.2 K) actually reduces the thermal noise power by a factor of about 70. In addition, both thermal and quantum noise can be reduced by reducing the bandwidth, that is, integrating over longer times, τ. This, however, requires stable conditions during the measurement time, τ. Unfortunately, however, other noise contributions may take over like shot noise and at low frequencies the so called 1/f noise.
Shot Noise
Shot noise originates from the discrete nature of the species carrying energy (e.g., electrons, photons). It was first discovered by Schottky [6] when studying the fluctuations of current in vacuum tubes. Shot noise is observed when the number of particles is small such that the statistical nature describing the occurrence of independent random events is described by the Poisson distribution. The Poisson distribution transforms into a normal (Gaussian) distribution as the number of particles increases. At low frequencies, shot noise is white, that is, the noise spectral density is independent of frequency and in contrast to the thermal noise also independent of temperature. The shot noise spectral density of an electrical current, Sel, at sufficiently low frequencies is given by
2.16
where I is the average current. Likewise, for a monochromatic photon flux, we have for the shot noise spectral density of photon flux, Sopt,
2.17
where hν is the photon energy and P the average power.
Low Frequency Noise (1/f Noise)
1/f noise (sometimes also called pink noise or flicker noise) occurs widely in nature but nevertheless might have quite different origin. More precisely, the relation between noise power spectral density and frequency often is given by
2.18
with β mostly close to 1. In contrast to thermal or quantum noise, the noise power of 1/f noise decreases with increasing frequency (by 3 dB per octave of frequency). Figure 2.3 shows, for example, the noise power spectral density as measured for a superconducting quantum interference device (SQUID) magnetometer versus frequency [7].
Figure 2.3 Noise power spectral density as measured for a SQUID magnetometer versus frequency.
(From...
