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In this paper, we present a model describing the modesplitting effect in a whispering-gallery (WG)-mode resonator. The WG mode is represented by an equivalent transmission-line ring resonator in which a localized perturbation is introduced. This perturbation induces a coupling between the two counter propagating waves, which, in turn, lifts the degeneracy of the WG modes. This simple model explains almost all the line shapes experimentally observed. It shows in a straightforward way how the coupling on the two twin modes are affected by changing the coupling probe and the perturbation angular positions. In that sense, it is useful to make the resonator adjustment easier. Moreover, the same approach can be used to simulate any microstrip ring resonator with localized perturbation. Index Terms--Microwave oscillator, ring resonator, ultra-stable oscillator, whispering-gallery (WG) mode.

The implementation of a microwave cryogenic sapphire oscillator (CSO) at the Laboratoire de Physique et Metrologie des Oscillateurs is described. The CSO compared to a hydrogen maser demonstrates frequency stability better than 3 X 10(super -14) at short term.

New measurements of dielectric properties of Lanthanum gallium silicate (Langasite or LGS) conducted with the whispering-gallery mode technique at microwave frequencies is calculated along with the first measurements of the two independent components of the imaginary part of the permittivity tensor. It appears LGS is a relatively high-loss dielectric material compared with sapphire or quartz.

Here, we study the paramagnetic ions behavior in presence of a strong microwave electromagnetic field sustained inside a cryogenic sapphire whispering gallery mode resonator. The high frequency measurement resolution that can be now achieved by comparing two Cryogenic Sapphire Oscillators (CSOs) permit to observe clearly the non-linearity of the resonator power sensitivity. These observations that, in turn, allow us to optimize the CSO operation are well explained by the electron spin resonance saturation of the paramagnetic impurities contained in the sapphire crystal. [ABSTRACT FROM AUTHOR]

The Cryogenic Sapphire Oscillator (CSO) is currently the best available technology that can provide a relative frequency stability better than 10−15 with integration times between 1 s and 10,000 s. But, the CSO remains a complex instrument that requires multiple loop controls to achieve the best frequency stability. The possibility to use the sapphire resonator in a self-sustained MASER oscillator presents an elegant alternative to the CSO. Here, sustaining the amplification is achieved through the interaction between a high-Q factor whispering gallery mode and the paramagnetic Fe3+ ions, which are present in small concentration in the sapphire crystal. The Fe3+ ion exhibits three energy states enabling to realize a self-sustaining solid-state maser. Although, this principle has been already experimentally demonstrated few years ago, its development as a truly usable ultra-stable source has not yet been completed, mainly due to the lack of control of the complex physical phenomena involved. This paper complements the previous theoretical work based on the rate equations model. Here we derive the full quantum equations describing the evolution of the Fe3+ ions inside the sapphire lattice and submitted to a pump and a maser signal. The influence of the ions concentration and spin-spin relaxation time will be pointed out. [ABSTRACT FROM AUTHOR]

Software-defined radio (SDR) provides stability, flexibility, and reconfigurability to radio frequency signal processing. Applied to oscillator characterization in the context of ultrastable clocks, stringent filtering requirements are defined by spurious signal or noise rejection needs. Since real-time radio frequency processing must be performed in a field-programmable array to meet timing constraints, we investigate optimization strategies to design filters meeting rejection characteristics while limiting the hardware resources required and keeping timing constraints within the targeted measurement bandwidths. The presented technique is applicable to scheduling any sequence of processing blocks characterized by a throughput, resource occupation and performance tabulated as a function of configuration characteristics, as is the case for filters with their coefficients and resolution yielding rejection and the number of multipliers. [ABSTRACT FROM AUTHOR]

This paper introduces the \Omega counter, a frequency counter—i.e., a frequency-to-digital converter—based on the linear regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the \Omega counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a system on chip, under test in our laboratory, and we compare the \Omega counter to the traditional \Pi and \Lambda counters. The LR exhibits the optimum rejection of white phase noise, superior to that of the \Pi and \Lambda for the \Pi counter, and to 1/\tau ^{3} for both the \Lambda $ and $\Omega $ counters. However, the $\Omega $ counter has the smallest possible variance, 1.25 dB smaller than that of the $\Lambda $ counter. The $\Omega $ counter finds a natural application in the measurement of the parabolic variance, described in the companion article in this Journal [vol. 63 no. 4 pp. 611–623, April 2016 (Special Issue on the 50th Anniversary of the Allan Variance), DOI 10.1109/TUFFC.2015.2499325]. [ABSTRACT FROM PUBLISHER]

This paper introduces the parabolic variance (PVAR), a wavelet variance similar to the Allan variance (AVAR), based on the linear regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the $\Omega$ frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and modified AVAR (MVAR). PVAR is good for long-term analysis because the wavelet spans over $2\tau$, the same as the AVAR wavelet, and good for short-term analysis because the response to white and flicker PM is $1/\tau^3$ and $1/\tau^2$, the same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition—or corner—where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift. [ABSTRACT FROM AUTHOR]

This article reports on the characterization of cryogenic sapphire oscillators (CSOs), and on the first test of a CSO in a real field installation, where ultimate frequency stability and continuous operation are critical issues, with no survey. Thanks to low-vibration liquid-He cryocooler design, Internet monitoring, and a significant effort of engineering, these oscillators could bridge the gap from an experiment to a fully reliable machine. The cryocooler needs scheduled maintenance every 2 years, which is usual for these devices. The direct comparison of two CSOs demonstrates a frequency stability of 5 × 10-16 for 30 s <= τ <= 300 s integration time, and 4.5 × 10-15 at 1 day (1 × 10-14 typical). Two prototypes are fully operational, codenamed ELISA and ULISS. ELISA has been permanently installed the new deep space antenna station of the European Space Agency in Malargüe, Argentina, in May 2012. ULISS is a transportable version of ELISA, modified to fit in a small van (8.5 m2 footprint). Installation requires a few hours manpower and 1 day of operation to attain full stability. ULISS, intended for off-site experiments and as a technology demonstrator, and has successfully completed two long-distance travels. [ABSTRACT FROM AUTHOR]

We present new results on a cryogenic solid-state maser frequency standard, which relies on the excitation of whispering gallery (WG) modes within a doped monocrystalline sapphire resonator (α-Al2O3). Included substitutively within the highest purity HEMEX-grade sapphire crystal lattice are Fe2+ impurities at a concentration of parts per million, an unavoidable result of the manufacturing process. Mass conversion of Fe2+ to Fe3+ ions was achieved by thermally annealing the sapphire in air. Above-threshold maser oscillation was then excited in the resonator at zero applied DC magnetic field by pumping high-Q WG modes coincident in frequency with the electron spin resonance (ESR) energy levels of the Fe3+ spin population. A 2 stage annealing process was undertaken for a sapphire resonator with exceptionally low Fe3+ concentration, resulting in an improvement of 6 orders of magnitude in output power for this particular crystal, and exceeding the previous best implementation of our scheme in another crystal by nearly 20 dB. This represents an output signal 7 orders of magnitude more powerful than a typical commercial hydrogen maser. At this power level, we estimate a limit on the frequency stability of order 1 x 10 -17/√τ due to the Schawlow-Townes fundamental thermal noise limit. [ABSTRACT FROM AUTHOR]

The phase noise of a quartz crystal resonator working at liquid helium temperatures is studied. Measurement methods and the device environment are explained. The phase noise is measured for different resonance modes, excitation levels, amount of operating time, device orientations in relation to the cryocooler vibration axis, and temperatures. Stability limits of a frequency source based on such devices are evaluated in the present measurement conditions. The sources of phase flicker and white noises are identified. Finally, the results are compared with previous works. [ABSTRACT FROM PUBLISHER]

This article reports on the long-term frequency stability characterization of a new type of cryogenic sapphire oscillator using an autonomous pulse-tube cryocooler as its cold source. This new design enables a relative frequency stability of better than 4.5 x 10-15 over one day of integration. To the best of our knowledge, this represents the best long-term frequency stability ever obtained with a signal source based on a macroscopic resonator. [ABSTRACT FROM PUBLISHER]

Abstract: The properties of degenerate whispering spherical modes within a fused silica and sapphire dielectric resonator have been investigated. However, departure from spherical symmetry occurs due to the support structure and anisotropy of the material. Finite element modeling is used to predicted the lifting of the frequency degeneracy, and is confirmed by experiment. Further optimization of the sapphire resonator to realize a Michelson–Morley experiment is discussed. [Copyright &y& Elsevier]

This paper introduces the Ω counter, a frequency counter-i.e., a frequency-to-digital converter-based on the linear regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the Ω counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a system on chip, under test in our laboratory, and we compare the Ω counter to the traditional Π and Λ counters. The LR exhibits the optimum rejection of white phase noise, superior to that of the Π and Λ counters. White noise is the major practical problem of wideband digital electronics, both in the instrument internal circuits and in the fast processes, which we may want to measure. With a measurement time τ , the variance is proportional to 1/τ(2) for the Π counter, and to 1/τ(3) for both the Λ and Ω counters. However, the Ω counter has the smallest possible variance, 1.25 dB smaller than that of the Λ counter. The Ω counter finds a natural application in the measurement of the parabolic variance, described in the companion article in this Journal [vol. 63 no. 4 pp. 611-623, April 2016 (Special Issue on the 50th Anniversary of the Allan Variance), DOI 10.1109/TUFFC.2015.2499325].

We report observations of the Schawlow-Townes noise limit in a cryogenic sapphire secondary frequency standard. The effect causes a fundamental limit to the frequency stability, and was measured through the novel excitation of a bimodal maser oscillation of a Whispering Gallery doublet at 12.04 GHz. The beat frequency of 10 kHz between the oscillations enabled a sensitive probe for this measurement of fractional frequency instability of 10(-14) tau(-1/2) with only 0.5 pW of output power.