Nyquist Formula Research Articles

Resolution limit in opto-digital systems revisited.

The resolution limit achievable with an optical system is a fundamental piece of information when characterizing its performance, mainly in case of microscopy imaging. Usually this information is given in the form of a distance, often expressed in microns, or in the form of a cutoff spatial frequency, often expressed in line pairs per mm. In modern imaging systems, where the final image is collected by pixelated digital cameras, the resolution limit is determined by the performance of both, the optical systems and the digital sensor. Usually, one of these factors is considered to be prevalent over the other for estimating the spatial resolution, leading to the global performance of the imaging system ruled by either the classical Abbe resolution limit, based on physical diffraction, or by the Nyquist resolution limit, based on the digital sensor features. This estimation fails significantly to predict the global performance of opto-digital imaging systems, like 3D microscopes, where none of the factors is negligible. In that case, which indeed is the most common, neither the Abbe formula nor the Nyquist formula provide by themselves a reliable prediction for the resolution limit. This is a serious drawback since systems designers often use those formulae as design input parameters. Aiming to overcome this lack, a simple mathematical expression obtained by finely articulating the Abbe and Nyquist formulas, to easily predict the spatial resolution limit of opto-digital imaging systems, is proposed here. The derived expression is tested experimentally, and shows to be valid in a broad range of opto-digital combinations.

Thermal Fluctuations and Electromagnetic Noise Spectra in Quantum Statistical Mechanics

We derive the thermal noise spectrum of the longitudinal and transverse electric field operator of a given wave vector starting from the quantum-statistical definitions and relate it to the frequency and wave vector dependent complex conductivity in a homogeneous, isotropic system of electromagnetic interacting charged particles in the frame of the non-relativistic QED. No additional assumptions except the validity of linear response are used in the proof. The Nyquist formula for vanishing frequency, as well as the noise spectral density of Callen-Welton follow as byproduct. Furthermore we discuss also the noise of the photon occupation numbers.

Thermal noise of hot carriers under natural sunlight

The efficiency of hot carrier solar cells strongly depends on the attainable carrier temperature in the absorber material. Measurements of hot carrier temperature under natural concentrated sunlight were performed at thin films of graphite, bismuth and PbTe, cooled through AlN ceramic substrate. The carrier temperature was assessed by magnitude of thermal noise voltage of charge carriers in the frequency range between 300 and 20,000 Hz according to Nyquist formula. Noise measurement equipment was calibrated in a range of sample impedances by using heated metal film resistors. We found that the carrier temperature in graphite films can exceed 600 °C under up to 15,000× suns and is much higher than the estimated lattice temperature in the film under the given cooling conditions. Secondary electron micrographs illustrate a good mechanical contact between the graphite film and the AlN substrate. We found also distinct heating of charge carriers in PbTe up to about 100 °C. Contrary to that, no measurable carrier heating above the lattice temperature was detected in bismuth films. Johnson noise thermometry demonstrates potential in identification of attractive candidate materials for absorbers in hot carrier solar cells.

Extended Nyquist formula for a resistance subject to a heat flow

The Nyquist formula quantifies the thermal noise driven fluctuations of voltage across a resistance in equilibrium. We deal here with the case of a resistance driven out of equilibrium by putting it in contact with two thermostats at different temperatures. We reach a non-equilibrium steady state where a heat flux is flowing through the resistance. Our measurements demonstrate that a simple extension of the Nyquist formula to the non uniform temperature field describes, with excellent precision, the thermal noise. For a metallic ohmic material, the fluctuations are actually equivalent to those of a resistance in equilibrium with a single thermostat at the mean temperature between the hot and cold sources.

Detrending and Other Features of Data Processing in the Measurements of Electrochemical Noise

Measurements of electrochemical noise in solid-state electrochemical cells with a heteropolycompound- based electrolyte are carried out. The noise power spectral density is calculated using various detrending methods. The impedance real component is calculated by means of the Nyquist formula. The results of the calculations are compared with the data obtained by the electrochemical impedance classical method.

The Evaluation of Noise Spectroscopy Tests

The paper discusses mathematical tools to evaluate novel noise spectroscopy based analysis and describes, via physical similarity, the mathematical models expressing the quantitative character of the modeled task. Using the Stefan–Boltzmann law, the authors indicate finding the spectral density of the radiated power of a hemisphere, and, for the selected frequency interval and temperature, they compare the simplified models with the expression of noise spectral density according to the Johnson–Nyquist formula or Nyquist’s expression of the function of spectral density based on a derivation of Planck’s law. The related measurements and evaluations, together with analyses of the noise spectroscopy of periodic resonant structures, are also outlined in the given context.

Temperature measurement using thermal noise conducting objects

Introduction. A noise method of temperature measurement is widely used in engineering and science. The disadvantage of noise methods is dependent measurement results of the resistance of the object.Main part. In conducting object is always present thermal noise. He called the velocity fluctuations of electric charge carriers. The paper shows the possibility of electrically conductive object temperature measurement by assessing their own thermal noise. The structure of the construction of the noise temperature of the meter and the algorithm of its work. The scheme includes the following elements: lead object, separate capacitors, transformer, broadband amplifiers, multiplier, low-pass filter and voltmeter. RMS noise voltage is determined by the Nyquist formula. The algorithm of the measuring system eliminates the effect of the resistance of the object on the measuring result. Limiting thermal noise band occurs in the resonant circuit. This eliminates the noise correlation own band amplifiers. The scheme allows for the measurement of low temperatures in the low-thermal noise.Conclusions. The independence of the measurement results of temperature resistance thermal noise source allows refusing sensors. Considered measuring the temperature comfortable in when placing sensors at the site difficult.

Zero Thermal Noise in Resistors at Zero Temperature

The bandwidth of transistors in logic devices approaches the quantum limit, where Johnson noise and associated error rates are supposed to be strongly enhanced. However, the related theory — asserting a temperature-independent quantum zero-point (ZP) contribution to Johnson noise, which dominates the quantum regime — is controversial and resolution of the controversy is essential to determine the real error rate and fundamental energy dissipation limits of logic gates in the quantum limit. The Callen–Welton formula (fluctuation–dissipation theorem) of voltage and current noise for a resistance is the sum of Nyquist’s classical Johnson noise equation and a quantum ZP term with a power density spectrum proportional to frequency and independent of temperature. The classical Johnson–Nyquist formula vanishes at the approach of zero temperature, but the quantum ZP term still predicts non-zero noise voltage and current. Here, we show that this noise cannot be reconciled with the Fermi–Dirac distribution, which defines the thermodynamics of electrons according to quantum-statistical physics. Consequently, Johnson noise must be nil at zero temperature, and non-zero noise found for certain experimental arrangements may be a measurement artifact, such as the one mentioned in Kleen’s uncertainty relation argument.

Zero-point term and quantum effects in the Johnson noise of resistors: a critical appraisal

There is a longstanding debate about the zero-point term in the Johnson noise voltage of a resistor. This term originates from a quantum-theoretical treatment of the fluctuation-dissipation theorem (FDT). Is the zero-point term really there, or is it only an experimental artifact, due to the uncertainty principle, for phase-sensitive amplifiers? Could it be removed by renormalization of theories? We discuss some historical measurement schemes that do not lead to the effect predicted by the FDT, and we analyse new features that emerge when the consequences of the zero-point term are measured via the mean energy and force in a capacitor shunting the resistor. If these measurements verify the existence of a zero-point term in the noise, then two types of perpetual motion machines can be constructed. Further investigation with the same approach shows that, in the quantum limit, the Johnson–Nyquist formula is also invalid under general conditions even though it is valid for a resistor-antenna system. Therefore we conclude that in a satisfactory quantum theory of the Johnson noise, the FDT must, as a minimum, include also the measurement system used to evaluate the observed quantities. Issues concerning the zero-point term may also have implications for phenomena in advanced nanotechnology.

Test of the fluctuation theorem for single-electron transport

Using time-resolved charge detection in a double quantum dot, we present an experimental test of the fluctuation theorem. The fluctuation theorem, a result from nonequilibrium statistical mechanics, quantifies the ratio of occurrence of fluctuations that drive a small system against the direction favored by the second law of thermodynamics. Here, these fluctuations take the form of single electrons flowing against the source–drain bias voltage across the double quantum dot. Our results, covering configurations close to as well as far from equilibrium, agree with the theoretical predictions, when the finite bandwidth of the charge detection is taken into account. In further measurements, we study a fluctuation relation that is a generalization of the Johnson–Nyquist formula and relates the second-order conductance to the voltage dependence of the noise. Current and noise can be determined with the time-resolved charge detection method. Our measurements confirm the fluctuation relation in the nonlinear transport regime of the double quantum dot.

A Simple Wide-Range Noise Thermometer With DC SQUID Readout for Operation Down to 10 mK

The design and performance of a compact and easy-to-build thermometer for the temperature range of 10 mK-4.2 K, which is wide enough to cover the entire temperature range of a typical dilution refrigerator, are described here. A commercial low-T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> dc superconducting quantum interference device (SQUID) is used to measure the thermal current noise of a resistor, i.e., an insulated copper wire, that scales linearly with the temperature, as expected from the Nyquist formula. Although, in principle, this thermometer is a primary thermometer, its performance was evaluated as a secondary thermometer. The relative precision of the temperature, which is independent of the temperature, was estimated at 0.4% in a measuring time of 130 s, in good agreement with the expectations. The accuracy evaluated from the comparison with other calibrated thermometers (i.e., superconducting reference point (SRP), cerium magnesium nitrate, and germanium) on different temperature ranges is better than 2% between 10 mK and 4.2 K. If only the SRP thermometer is considered for the comparison in the temperature range of 20-207 mK, the accuracy improves to 1%. The calibrations performed with a single fixed point have a typical accuracy value of about 1%.

On the quantum Nyquist formula

Yu.L. Klimontovich’s assertion on the incorrectness of the quantum Nyquist formula is discussed. It is shown that, in quantum physics, Rytov’s fluctuation formula, the Casimir force theory as treated by E.M. Lifshitz, and the quantum Nyquist formula are all incorrect, as they are united by a common basis.

Current-sensing noise thermometry from [formula omitted] to below [formula omitted] using a DC SQUID preamplifier

We are using a low- T c DC SQUID to perform current-sensing noise thermometry, by measuring the thermal noise currents in a copper resistor. The temperature is obtained from the Nyquist formula. This is a practical thermometer for use from 4.2 K to below 1 mK , with a percentage precision independent of temperature. Using a 0.34 mΩ resistor, the thermometer had an amplifier noise temperature T N of 8 μK . A precision of 1.5% was obtained in 200 s . The thermometer was in good agreement with the PLTS-2000 3 He melting curve scale down to 4.5 mK . We have cooled the thermometer successfully below 1 mK , achieving a minimum electron temperature of 300 μK .

Analysis of noise in energy-dispersive spectrometers using time-domain methods

This paper presents an integrated time domain approach to the optimization of the signal-to-noise ratio in all spectrometer systems that contain a detector that converts incoming quanta of radiation into electrical pulse signals that are amplified and shaped by an electronic pulse shaper. It allows analysis of normal passive pulse shapers as well as time-variant systems where switching of shaping elements occurs in synchronism with the signal. It also deals comfortably with microcalorimeters (sometimes referred to as bolometers), where noise-determining elements, such as the temperature-sensing element's resistance and temperature, change with time in the presence of a signal. As part of the purely time-domain approach, a new method of calculating the Johnson noise in resistors using only the statistics of electron motion is presented. The result is a time-domain analog of the Nyquist formula.

Effect of thermal and quantum fluctuations on superfluid density in layered superconductors

We calculate suppression of inter- and intra-layer superconducting currents due to equilibrium phase fluctuations and find that, in contrast to a recent prediction, the effect of thermal fluctuations cannot account for linear temperature dependence of the superfluid density in high- T c superconductors at low temperatures. Quantum fluctuations are found to dominate over thermal fluctuations at low temperatures due to hardening of their spectrum caused by the Josephson plasma resonance. Near T c sizeable thermal fluctuations are found to suppress the critical current in the stack direction stronger than in the direction along the layers. Fluctuations of quasi-particle branch imbalance make the spectral density of voltage fluctuations at small frequencies non-zero, in contrast to what may be expected from a naive interpretation of Nyquist formula.

Effect of equilibrium fluctuations on superfluid density in layered superconductors.

Using a phenomenological approach based on the fluctuation-dissipation theorem we calculate suppression of superconducting currents due to phase fluctuations and find that, in contrast to a recent prediction, the effect of thermal fluctuations cannot account for linear temperature dependence of the superfluid density in high-T(c) superconductors at low temperatures. Quantum fluctuations are found to dominate over thermal fluctuations at low temperatures. Near T(c) sizable thermal fluctuations are found to suppress the critical current in the stack direction stronger than in the direction along the layers. The spectral density of voltage fluctuations at small frequencies is nonzero, in contrast to what may be expected from a naive interpretation of the Nyquist formula.

Current sensing noise thermometry using a low Tc DC SQUIDpreamplifier

We describe here the design and performance of a current sensingnoise thermometer using a low Tc DC SQUID as the front endamplifier. The DC SQUID is used to measure the thermal noise current in aresistor and the temperature is then obtained from the Nyquist formula. Thethermometer is fast, absolute and precise and is usable over a widetemperature range below 4.2 K, in principle down to well below 1 mK. Theexcellent energy sensitivity of the DC SQUID, operated at fixed temperature,enables the use of a relatively large noise resistor, in the mΩ range.This requires relatively short averaging times when measuring the spectrum ofnoise fluctuations. We have shown that it is possible to determine absolutetemperature with a precision of 1% in a measuring time of 10 seconds withan amplifier noise temperature, TN, of the order of 30 µK,and to an accuracy better than 0.3%. The percentage precision isindependent of temperature for temperatures much greater than TN.Our method of heat sinking the noise resistor ensures proper cooling of theelectrons. We incorporate a fixed point device for checking the gaincalibration. We have cooled the thermometer successfully to below 1 mK,achieving a minimum electron temperature of 300 µK. We presentthe results of a preliminary comparison with a 3He melting curvethermometer (MCT) above 4.5 mK, and with a platinum NMR thermometer down tothe lowest temperatures.

A thermodynamic noise model for nonlinear resistors

A Gaussian white noise model for passive nonlinear resistors, valid in the linear quadratic current-voltage (I-V) approximation at zero bias, is suggested. Although approximative, the model, which has originally been developed by R.L. Stratonovich, is thermodynamically well founded. The model is applied to the shot noise of exponential diodes and tunnel junctions as well as to the thermal noise of JFET's and MOSFET's. Within the range of the linear-quadratic approximation, the results are in full agreement with the well-known device specific standard low frequency thermal and shot noise models for these devices. The model can he considered as a linear-quadratic short circuit noise current generalization of Nyquist's formula.

On the quantum limit for resolution in force measurement using an optical displacement transducer

On the example of the simplest optical transducer it is shown that in continuous coordinate measurement, back action does not limit the value of the minimum registered force if the radiative friction introduced by the device is not taken into account. It means that there is no standard quantum limit (SQL) for the resolution in the measurements of force. With radiative friction taken into account, the minimal uncertainty in force ΔF min ⋍ξΔF SQL (ω F /ω 0) 1 4 is calculated, where ω F i s the frequency of force, ω 0 is optical frequency and ξ a constant of the order of unity, achievable at optimum pump power. For a mechanical resonator with intrinsic friction larger than the radiative one, the minimum error in the force measurement (at optimum pump power) is described by the quantum Nyquist formula with zero-point fluctuations energy ħΩ/2 substituted by ħΩ. In the reflected wave one has to measure not the phase but the quadrature amplitude in a modified homodyne scheme. It is also shown that the pump can be in a coherent state: no squeezed, photon number or any nonclassical states are required.

Corpuscular theory of intensity noise with gain compression.

When light from a laser is fully absorbed by an ideal detector, the detected current exhibits a fluctuation called here ``photonic noise.'' The spectral density of Iintensity noiseR, defined as the difference of the photonic-noise spectral density and a term corresponding to the shot-noise level, is negative for sub-Poissonian statistics. The usefulness of the relative-intensity-noise concept is that it is independent of any linear attenuation. A simple circuit theory of intensity noise based only on energy conservation and the Nyquist formula (zero-point fluctuation) leads to expressions of the spectral densities that agree with quantum theory even for sub-Poissonian photon statistics. When the optical gain and loss are frequency independent, the circuit theory reduces to a corpuscular theory that keeps track of the time rates of change of electron and photon numbers treated as continuous variables. Consideration is given to laser diodes in which the rate of electron-photon conversion depends nonlinearly on both the carrier and photon densities. The cross-spectral density between electrical-voltage and relative photonic fluctuations is independent of internal or external optical losses. Standard rate equations are inaccurate in the case of gain compression. Very general yet simple formulas for intensity noise are applied to room-temperature GaAs laser diodes, using recently calculated optical parameters.