A theoretical development is presented which results in a relationship between the expectation value of the standard deviation of the frequency fluctuations for any finite number of data samples and the infinite time average value of the standard deviation, which provides an invariant measure of an important quality factor of a frequency standard. A practical and straightforward method of determining the power spectral density of the frequency fluctuations from the variance of the frequency fluctuations, the sampling time, the number of samples taken, and the dependence on system bandwidth is also developed. Additional insight is also given into some of the problems that arise from the presence of "flicker noise" (spectrum proportional to |ω|-1) modulation of the frequency of an oscillator. The theory is applied in classifying the types of noise on the signals of frequency standards made available at NBS, Boulder Laboratories, such as: masers (both H and N15H 3 ), the cesium beam frequency standard employed as the U. S. Frequency Standard, and rubidium gas cells. "Flicker noise" frequency modulation was not observed on the signals of masers for sampling times ranging from 0.1 second to 4 hours. In a comparison between the NBS hydrogen maser and the NBS III cesium beam, uncorrelated random noise was observed on the frequency fluctuations for sampling times extending to 4 hours; the fractional standard deviations of the frequency fluctuations were as low as 5 parts in 1014.

Statistics of atomic frequency standards

A general model is introduced which is capable of making accurate, quantitative predictions about the phase noise of different types of electrical oscillators by acknowledging the true periodically time-varying nature of all oscillators. This new approach also elucidates several previously unknown design criteria for reducing close-in phase noise by identifying the mechanisms by which intrinsic device noise and external noise sources contribute to the total phase noise. In particular, it explains the details of how 1/f noise in a device upconverts into close-in phase noise and identifies methods to suppress this upconversion. The theory also naturally accommodates cyclostationary noise sources, leading to additional important design insights. The model reduces to previously available phase noise models as special cases. Excellent agreement among theory, simulations, and measurements is observed.

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A general theory of phase noise in electrical oscillators

We describe a novel oscillator that converts continuous light energy into stable and spectrally pure microwave signals. This optoelectronic microwave oscillator consists of a pump laser and a feedback circuit including an intensity modulator, an optical fiber delay line, a photodetector, an amplifier, and a filter. We develop a quasi-linear theory and obtain expressions for the threshold condition, the amplitude, the frequency, the line width, and the spectral power density of the oscillation. We also present experimental data to compare with the theoretical results. Our findings indicate that the optoelectronic microwave oscillator can generate ultrastable, spectrally pure microwave reference signals up to 75 GHz with a phase noise lower than -140 dBc/Hz at 10 kHz.

Optoelectronic microwave oscillator

Consider a signal generator whose instantaneous output voltage V(t) may be written as V(t) = [V 0 + ??(t)] sin [2??v 0 t + s(t)] where V 0 and v 0 are the nominal amplitude and frequency, respectively, of the output. Provided that ??(t) and ??(t) = (d??/(dt) are sufficiently small for all time t, one may define the fractional instantaneous frequency deviation from nominal by the relation y(t) - ??(t)/2??v o A proposed definition for the measure of frequency stability is the spectral density S y (f) of the function y(t) where the spectrum is considered to be one sided on a per hertz basis. An alternative definition for the measure of stability is the infinite time average of the sample variance of two adjacent averages of y(t); that is, if y k = 1/t ??? tk+r = y(t k ) y(t) dt where ?? is the averaging period, t k+1 = t k + T, k = 0, 1, 2 ..., t 0 is arbitrary, and T is the time interval between the beginnings of two successive measurements of average frequency; then the second measure of stability is ?? y 2(??) ??? (y k+1 - y k )2/2 where denotes infinite time average and where T = ??. In practice, data records are of finite length and the infinite time averages implied in the definitions are normally not available; thus estimates for the two measures must be used. Estimates of S y (f) would be obtained from suitable averages either in the time domain or the frequency domain.

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Characterization of Frequency Stability

Micromotion of ions in Paul traps has several adverse effects, including alterations of atomic transition line shapes, significant second-order Doppler shifts in high-accuracy studies, and limited confinement time in the absence of cooling. The ac electric field that causes the micromotion may also induce significant Stark shifts in atomic transitions. We describe three methods of detecting micromotion. The first relies on the change of the average ion position as the trap potentials are changed. The second monitors the amplitude of the sidebands of a narrow atomic transition, caused by the first-order Doppler shift due to the micromotion. The last technique detects the Doppler shift induced modulation of the fluorescence rate of a broad atomic transition. We discuss the detection sensitivity of each method to Doppler and Stark shifts, and show experimental results using the last technique.

Minimization of ion micromotion in a Paul trap

Precision quartz oscillators have three main sources of noise contributing to frequency fluctuations: thermal noise in the oscillator, additive noise contributed by auxiliary circuitry such as AGC, etc., and fluctuations in the quartz frequency itself as well as in the reactive elements associated with the crystal, leading to an f-1type of power spectral density in frequency fluctuations. Masers are influenced by the first two types of noise, and probably also by the third. The influence of these sources of noise on frequency fluctuation vs. averaging time measurements is discussed. The f-1-spectral density leads to results that depend on the length of time over which the measurements are made. An analysis of the effects of finite observation time is given. The characteristics of both passive and active atomic standards using a servo-controlled oscillator are discussed. The choice of servo time constant influences the frequency fluctuations observed as a function of averaging time and should be chosen for best performance with a given quartz oscillator and atomic reference. The conventional methods of handling random signals, i.e., variances, autocorrelation, and spectral densities, are applied to the special case of frequency and phase fluctuations in oscillators, in order to obtain meaningful criteria for specifying oscillator frequency stability. The interrelations between these specifications are developed in the course of the paper.

Some aspects of the theory and measurement of frequency fluctuations in frequency standards

The local clocks within each node of a loosely distributed network system may be synchronized and syntonized by any of the nodes in the network. Each of the nodes contains a time packet detector (TPD) that detects and recognizes timing data packets and produces a recognition signal. This signal may be used to cause an action in the detecting node, in particular the node may contain a time service block (TSB) containing a local clock that may be used to record the time of the recognition signal. The recorded times may be used as the basis for correcting the local clocks of the nodes in the system. Transfer devices such as gateways, bridges, and routers may include TSBs and TPDs to allow for correction for the transit time of time packets through the transfer device. The TPD is normally connected within the media access device at the point of final encoding for transmission or recovery of the clock and data for receiving where the time jitter will be at a minimum.

Method for recognizing events and synchronizing clocks

The largest systematic uncertainty in the performance of atomic frequency standards using a cloud of ions stored in an rf quadrupole trap is the second-order Doppler shift which depends on ion temperature and trapping parameters. This paper presents evidence that cooling the ions by collisions with atoms of a background gas light compared to the ions results in the condensation of the ions into a cloud of almost uniform density determined by space charge versus potential well forces. In this condition the second-order Doppler shift is simple to calculate and is found to depend only on readily measured characteristics of the ion cloud. This along with already observed good signal-to-noise ratio shows that the frequency standard we have constructed using the hyperfine splitting of singly ionized199Hg, with helium cooling can have an order of magnitude better performance in accuracy, stability, and reproducibility than presently available commercial cesium beam standards.

Thermalization of199Hg ion macromotion by a light background gas in an RF quadrupole trap

The second-order Doppler shift is an important source of systematic error in rf quadrupole trapped ion frequency standards. This shift can be reduced by cooling the secular motion of the ion cloud with a light background gas at low pressure. Using a thermalized ion cloud model, it is possible to relate the Doppler shift to the temperature of the ion cloud. It is shown that, in practice, the measured frequency of the first-order Doppler sidebands can be used to determine the ion cloud temperature.

Leonard S. Cutler

Papers

Some aspects of the theory and measurement of frequency fluctuations in frequency standards

Method for recognizing events and synchronizing clocks

Thermalization of199Hg ion macromotion by a light background gas in an RF quadrupole trap

Doppler effects due to thermal macromotion of ions in an rf quadrupole trap

Draft revision of IEEE STD 1139-1988 standard definitions of physical quantities for fundamental frequency and time metrology — random instabilities