Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovich dilemma

We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models

Study of voltage controlled oscillator based analog-to-digital converter

A voltage controlled oscillator (VCO) based analog-to-digital converter (ADC) is a time based architecture with a first-order noise-shaping property, which can be implemented using a VCO and digital circuits. This thesis analyzes the performance of VCO-based ADCs in the presence of non idealities such as jitter, nonlinearity, mismatch, and the metastability of D flip-flops. Based on this analysis, design criteria for determining parameters for VCO-based ADCs are described. Further, the study involves the use of VCO based Dual-slope A/D converter and its behaviour under different input voltage level. Graph is plotted between output voltages of the integrator vs. time. Digital circuits like a bit-counter and logic circuits are used for operation mode. A normal VCO model is also done in MATLAB-simulink environment and studied under variable input frequency and corresponding output plots are view

Fundamental Carrier-Envelope Phase Noise Limitations during Pulse Formation and Detection

The difference between the positions of the maximum peak of the carrier wave of a laser pulse and the maximum of its intensity envelope is termed carrier-envelope phase (CEP). In the last decades, the control and stabilization of this parameter has greatly improved, which enables many applications in research fields that rely on CEP-stable pulses such as attosecond science and optical frequency metrology. Further progress in these fields depends strongly on minimizing the CEP noise that restricts stabilization performance. While the CEP of most high repetition-rate low-energy laser oscillators has been stabilized to a remarkable precision, some types of oscillators show extensive noise that inhibits precise stabilization. The CEP stabilization performance of low repetition-rate high peak-power amplified laser systems also remains limited by noise, which is believed to stem mainly from the CEP detection process. In this thesis, the origins of the CEP noise within four oscillators as well as the noise induced by the measurement of the CEP of amplified pulses are investigated. In the first part, the properties of the CEP noise of one Ti:sapphire oscillator and three different fiber oscillators are extracted by analyzing the unstabilized CEP traces by means of time-resolved correlation analysis of carrier-envelope amplitude and phase noise as well as by methods that reveal the underlying statistical noise properties. In the second part, investigations into the origin of CEP noise induced by the measurement of the CEP of amplified pulses are conducted by comparing several different CEP detection designs that are based on f -2 f interferometry. These detection setups differ in the employed sources of spectral broadening as well as frequency doubling media, both necessary steps to measure the CEP. The results in both parts of this thesis show that white quantum noise dominates most CEP measurements. In one particular fiber oscillator, the strong white noise is found to be a result of a correlating mechanism within the employed SESAM. During amplifier CEP detection, the CEP noise is found to be originating only to a marginal degree from the number of photons that are detected during the measurement, which excludes shot noise as a limiting source. Instead, the analysis reveals that the origin of the observed strong white noise can be interpreted as a loss of coherence during detection. This type of coherence is termed here intra-pulse coherence and describes the phase transfer within f -2 f interferometry. Its degradation is a result of amplitude-to-phase coupling during the spectral broadening process that leads to pulse-to-pulse fluctuations of the phases at the edges of the extended spectrum. Numerical simulations support the concept of intra-pulse coherence degradation and show that the degradation is substantially stronger during plasma-driven spectral broadening as compared to self-phase modulation-dominated spectral broadening. This difference in degradation also explains the much stronger CEP noise typically observed in amplified systems as compared to oscillators, as the former typically rely on filamentation-based and hence plasma-dominated spectral broadening for CEP detection. The concept of intra-pulse coherence constitutes a novel measure to assess the suitability of a spectral broadening mechanism for application in active as well as in passive CEP stabilization schemes and provides new strategies to reduce the impact of the CEP detection on the overall stabilization performance of most lasers.Diese Arbeit beschäftigt sich mit der Identifizierung und Minimierung fundamentaler Rauschquellen, die zu einer Limitierung des erreichbaren Carrier-Envelope Phasen (CEP) Jitters führen. Die Carrier-Envelope Phase beschreibt die Differenz zwischen dem Maximum der Trägerwelle und dem Scheitelpunkt der Intensitätseinhüllenden. In den letzten Jahrzehnten hat sich die Kontrolle und Stabilisierung der CEP deutlich verbessert, was zu einem schnellen Fortschritt in Forschungsfeldern geführt hat, bei denen CEP-stabile Pulse notwendig sind. Diese Forschungsfelder umfassen die Attosekundenforschung und optische Frequenzmetrologie. Weitere Entwicklungen in diesen Feldern hängt stark von der Minimierung von CEP Rauschen ab, welches die CEP Stabilisierung stark beeinträchtigt. Obwohl die CEP der Pulse der meisten Laseroszillatoren mit hohen Repetitionsraten äußerst genau stabilisiert werden kann, existieren einige Laseroszillatoren bei denen starke Rauschquellen eine Stabilisierung verhindern oder stark einschränken. Des Weiteren zeigen vor Allem verstärkte System mit niedrigen Repetitionsraten und hohen Spitzenleistungen eine Beschränkung der CEP Stabilisierung aufgrund von Rauschen, dass vermutlich zum großen Teil durch den Detektionsprozess entsteht. In dieser Arbeit ist der Ursprung von CEP Rauschen in vier unterschiedlichen Laseroszillatoren sowie während der Detektion der CEP von verstärkten Systemen untersucht worden. Im ersten Teil wurden die Eigenschaften des CEP Rauschens eines Ti:Saphir-basierten Oszillators und drei verschiedener Faserlaser analysiert. Hierzu wurde das Rauschen unter anderem mittels zeitaufgelöster Korrelationsanalyse von Carrier-Envelope Amplituden- und Phasenrauschen sowie mittels Methoden, die die statistischen Eigenschaften des Rauschens offenlegen, analysiert. Im zweiten Teil der Arbeit wurde das Rauschen untersucht, welches durch den Messprozess der CEP von verstärkten Pulsen mittels f -2 f Interferometrie entsteht. Experimentell wurden hierzu vier unterschiedliche Detektionsanordnungen verwendet, die sich durch die Nutzung unterschiedlicher nichtlinearer Prozesse zum Erzeugen der spektralen Verbreiterung sowie zur Erzeugung der zweiten Harmonischen unterscheiden. Die Ergebnisse in beiden Teilen der Arbeit zeigen dominierendes weißes Quantenrauschen in den meisten CEP Messungen. In einem bestimmten Faserlaser, in dem besonders starkes weißes Rauschen vorlag, konnte der Ursprung einerWechselwirkung innerhalb des verwendeten halbleiterbasierten sättigbaren Absorbers zugeordnet werden. Bei der Detektion der CEP bei verstärkten Systemen wurde hingegen gezeigt, dass niedrige Photonenzahlen und damit Schrotrauschen nur zum kleinen Teil für die starken weißen Rauschanteile verantwortlich gemacht werden kann. Stattdessen kann die Ursache des starken Rauschens einem Verlust von Kohärenz zugeordnet werden. Diese Art von Kohärenz ist hier mit intra-Puls Kohärenz bezeichnet und beschreibt den Phasentransfer innerhalb der Detektion mittels f -2 f Interferometrie. Der Verlust von intra-Puls Kohärenz ist eine Folge von Amplituden-zu-Phasen Koppelung während der spektralen Verbreiterung. Von Puls zu Puls führt dies zu Fluktuationen der Phase an beiden Rändern der erzeugten spektralen Verbreiterung. Numerische Simulationen unterstützen das Konzept der intra-Puls Kohärenz und zeigen auf, dass die Degradation bedeutend stärker bei plasmadominierten Prozessen ausfällt als im Vergleich zu spektraler Verbreiterung mittels Selbstphasenmodulation. Dieser unterschiedlich starke Verlust der intra-Puls Kohärenz erklärt das deutlich höhere Rauschniveau in verstärkten Systemen im Vergleich zu Oszillatoren, da verstärkte Systeme plasmadominierte Prozesse zur spektralen Verbreiterung nutzen. Das Konzept der intra-Puls Kohärenz stellt ein neues Maß zur Einschätzung einer Methode zur spektralen Verbreiterung für eine bestimmte Anwendung dar, die sowohl in aktiven sowie passiven CEP Stabilisierungen von Lasern eine Rolle spielt. Es ermöglicht somit neue Strategien, um den Einfluss der Detektion auf die CEP Stabilisierung der meisten Laser zu senken

An analytical formulation for phase noise in MEMS oscillators.

In recent years, there has been much interest in the design of low-noise MEMS oscillators. This paper presents a new analytical formulation for noise in a MEMS oscillator encompassing essential resonator and amplifier nonlinearities. The analytical expression for oscillator noise is derived by solving a second-order nonlinear stochastic differential equation. This approach is applied to noise modeling of an electrostatically addressed MEMS resonator-based square-wave oscillator in which the resonator and oscillator circuit nonlinearities are integrated into a single modeling framework. By considering the resulting amplitude and phase relations, we derive additional noise terms resulting from resonator nonlinearities. The phase diffusion of an oscillator is studied and the phase diffusion coefficient is proposed as a metric for noise optimization. The proposed nonlinear phase noise model provides analytical insight into the underlying physics and a pathway toward the design optimization for low-noise MEMS oscillators.The authors would like to thank the UK-Indian Education and Research Initiative (grant SA06-250) and the Cambridge Trusts for funding support.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TUFFC.2014.00651

RF MEMS reference oscillators platform for wireless communications

A complete platform for RF MEMS reference oscillator is built to replace bulky quartz from mobile devices, thus reducing size and cost. The design targets LTE transceivers. A low phase noise 76.8 MHz reference oscillator is designed using material temperature compensated AlN-on-silicon resonator. The thesis proposes a system combining piezoelectric resonator with low loading CMOS cross coupled series resonance oscillator to reach state-of-the-art LTE phase noise specifications. The designed resonator is a two port fundamental width extensional mode resonator. The resonator characterized by high unloaded quality factor in vacuum is designed with low temperature coefficient of frequency (TCF) using as compensation material which enhances the TCF from - 3000 ppm to 105 ppm across temperature ranges of -40˚C to 85˚C. By using a series resonant CMOS oscillator, phase noise of -123 dBc/Hz at 1 kHz, and -162 dBc/Hz at 1MHz offset is achieved. The oscillator’s integrated RMS jitter is 106 fs (10 kHz–20 MHz), consuming 850 μA, with startup time is 250μs, achieving a Figure-of-merit (FOM) of 216 dB. Electronic frequency compensation is presented to further enhance the frequency stability of the oscillator. Initial frequency offset of 8000 ppm and temperature drift errors are combined and further addressed electronically. A simple digital compensation circuitry generates a compensation word as an input to 21 bit MASH 1 -1-1 sigma delta modulator incorporated in RF LTE fractional N-PLL for frequency compensation. Temperature is sensed using low power BJT band-gap front end circuitry with 12 bit temperature to digital converter characterized by a resolution of 0.075˚C. The smart temperature sensor consumes only 4.6 μA. 700 MHz band LTE signal proved to have the stringent phase noise and frequency resolution specifications among all LTE bands. For this band, the achieved jitter value is 1.29 ps and the output frequency stability is 0.5 ppm over temperature ranges from -40˚C to 85˚C. The system is built on 32nm CMOS technology using 1.8V IO device

TDRSS/user satellite timing study

A timing analysis for data readout through the Tracking and Data Relay Satellite System (TDRSS) was presented. Various time tagging approaches were considered and the resulting accuracies delineated. The TDRSS was also defined and described in detail

Design and Implementation of Silicon-Based MEMS Resonators for Application in Ultra Stable High Frequency Oscillators

The focus of this work is to design and implement resonators for ultra-stable high-frequency ( \u3e 100MHz) silicon-based MEMS oscillators. Specifically, two novel types of resonators are introduced that push the performance of silicon-based MEMS resonators to new limits. Thin film Piezoelectric-on-Silicon (TPoS) resonators have been shown to be suitable for oscillator applications due to their combined high quality factor, coupling efficiency, power handling and doping-dependent temperature-frequency behavior. This thesis is an attempt to utilize the TPoS platform and optimize it for extremely stable high-frequency oscillator applications. To achieve the said objective, two main research venues are explored. Firstly, quality factor is systematically studied and anisotropy of single crystalline silicon (SCS) is exploited to enable high-quality factor side-supported radial-mode (aka breathing mode) TPoS disc resonators through minimization of anchor-loss. It is then experimentally demonstrated that in TPoS disc resonators with tethers aligned to [100], unloaded quality factor improves from ~450 for the second harmonic mode at 43 MHz to ~11,500 for the eighth harmonic mode at 196 MHz. Secondly, thickness quasi-Lamé modes are studied and demonstrated in TPoS resonators for the first time. It is shown that thickness quasi-Lamé modes (TQLM) could be efficiently excited in silicon with very high quality factor (Q). A quality factor of 23.2 k is measured in vacuum at 185 MHz for a fundamental TQLM-TPoS resonators designed within a circular acoustic isolation frame. Quality factor of 12.6 k and 6 k are also measured for the second- and third- harmonic TQLM TPoS resonators at 366 MHz and 555 MHz respectively. Turn-over temperatures between 40 °C to 125 °C are also designed and measured for TQLM TPoS resonators fabricated on degenerately N-doped silicon substrates. The reported extremely high quality factor, very low motional resistance, and tunable turn-over temperatures \u3e 80 °C make these resonators a great candidate for ultra-stable oven-controlled high-frequency MEMS oscillators