In theoretical physics, the word “oscillator” refers to a physical object or quantity oscillating sinusoidally – or at least periodically – for a long time, ideally forever, without losing its initial energy. An example of an oscillator is the classical atom, where the electrons rotate steadily around the nucleus. Conversely, in experimental science the word “oscillator” stands for an artifact that delivers a periodic signal, powered by a suitable source of energy. In this book we will always be referring to the artifact. Examples are the hydrogen maser, the magnetron of a microwave oven, and the swing wheel of a luxury wrist watch. Strictly, a “clock” consists of an oscillator followed by a gearbox that counts the number of cycles and the fraction thereof. In digital electronics, the oscillator that sets the timing of a system is also referred to as the clock. Sometimes the term “atomic clock” is improperly used to mean an oscillator stabilized to an atomic transition, because this type of oscillator is most often used for timekeeping.

A large part of this book is about the “precision” of the oscillator frequency and about the mechanisms of frequency and phase fluctuations. Before tackling the main subject, we have to go through the technical language behind the word “precision,” and present some elementary mathematical tools used to describe the frequency and phase fluctuations.

INTRODUCTION

We asserted in Chapter 15 that tuned oscillators produce outputs with higher spectral purity than relaxation oscillators. One straightforward reason is simply that a high-Q resonator attenuates spectral components removed from the center frequency. As a consequence, distortion is suppressed, and the waveform of a well-designed tuned oscillator is typically sinusoidal to an excellent approximation.

In addition to suppressing distortion products, a resonator also attenuates spectral components contributed by sources such as the thermal noise associated with finite resonator Q, or by the active element(s) present in all oscillators. Because amplitude fluctuations are usually greatly attenuated as a result of the amplitude stabilization mechanisms present in every practical oscillator, phase noise generally dominates – at least at frequencies not far removed from the carrier. Thus, even though it is possible to design oscillators in which amplitude noise is significant, we focus primarily on phase noise here. We show later that a simple modification of the theory allows for accommodation of amplitude noise as well, permitting the accurate computation of output spectrum at frequencies well removed from the carrier.

Aside from aesthetics, the reason we care about phase noise is to minimize the problem of reciprocal mixing. If a superheterodyne receiver's local oscillator is completely noise-free, then two closely spaced RF signals will simply translate downward in frequency together. However, the local oscillator spectrum is not an impulse and so, to be realistic, we must evaluate the consequences of an impure LO spectrum.

A study of phase noise

In this section we look into how phase noise arises from noise currents in an oscillator, using an analytic power series model. An analysis is conducted based on this power series model, and this is used to predict the phase noise of an oscillator. This circuit is then made to oscillate with transient simulation and is studied for its phase noise performance with the harmonic balance method using a proprietary simulator (ADS). The results are then compared. The basis of this study will be a half circuit test bench of a Colpitts oscillator as in Figure 10.14(a) and Figure 10.41 of Chapter 10, but with a HBT instead of a MOSFET.

The topic of phase noise was introduced in Section 10.1.4 of Chapter 10. Equation (10.15) simply assumed a phase noise existing at a frequency offset from the fundamental oscillation frequency, but does not explain how this phase noise arises from real physical noise current sources (e.g. resistors, lossy inductors, transistor shot noise, etc.) present inside the oscillator circuit. The latter is studied in more detail in this Appendix using a one-port equivalent circuit for the oscillator, as shown in Figure A10.1.

The inductor, Lpt , is assumed in parallel with a resistor Rpt, at the base input of the transistor. (An equivalent series L-R representation is also possible.) One end of this inductor is placed at the desired DC base bias voltage of the transistor. The value of Rpt is calculated based on the assumed Q of an actual linear inductor, at the oscillation frequency. The resistor Rpt is shown inside the “one-port” as illustrated in Figure A10.1. When this assembly is in steady-state oscillation, the input to the one-port must by definition be purely capacitive because the net negative resistance created inside the oscillator must balance any sources of positive resistance. Note that in the Colpitts oscillator the transistor inside the one-port does not have its emitter connected to ground, so the subsequent analysis of the one-port is of the prototype structure of this oscillator, not just of the proprietary device models included in it.

Quantum cascade lasers (QCL) can be powerful testing grounds of the fundamental physical parameters determined by their quantum nature. In this chapter we describe a set of experimental techniques to explore the linewidth, frequency and phase stability of far-infrared QCLs. By performing noise measurements with unprecedented sensitivity levels, we highlight the key role of gain medium engineering and demonstrate that properly designed semiconductor-heterostructure lasers can unveil the mechanisms underlying the laser-intrinsic phase noise, revealing the link between device properties and the quantum-limited linewidth. We discuss phase-locking of THz QCL to a free-space comb generated in a LiNbO3 waveguide, and present phase and frequency control of miniaturized QCL frequency combs. This work paves the way to novel metrological-grade THz applications, including high-resolution spectroscopy, manipulation of cold molecules, astronomy and quantum technologies. The physical processes and dynamics presented here open groundbreaking perspectives for the development of quantum sensors, quantum imaging devices and q-bits made by entangled teeth for photonic-based quantum computation.

Introduction

The evaluation of the oscillator phase noise is a classical issue. Some of the fundamental papers date back to the 1960s [1–4] and recently the topic has received fresh attention as full integration of RF systems has become the focus of microelectronic design. Designers of VCOs had to rely on noise models either empirically explained or based on highly questionable linear small-signal analysis. Since clear guidelines for circuit optimization were lacking, design was mainly based on a trial-and-error approach. Only at the end of the 1990s was a deeper insight in VCO noise analysis gained. Two frameworks were proposed: one working in the time domain, [5–8] the other in the frequency domain. [9–11] They both succeeded in providing the first quantitative guidelines to noise optimization, linking phase-noise performance to the transfer of the noise sources in the circuit. This grounding is essential for later figuring out proper options and modifications of the circuit topology.

Accurate circuit-level simulators, which have been developed meanwhile, [12–14] simplify and speed up the proper tuning of design parameters and the noise performance evaluation in every operating condition.

This chapter is devoted to describing time-domain and frequency-domain methods for the oscillator noise analysis. Some examples of phase noise calculation based on these theoretical frameworks and verified against the circuit-level simulator results are also shown.

Linear and time-invariant model

The spectrum of real oscillators is far from being a δ-like function at the oscillation frequency ω0.

The basic delay-line oscillator, shown in Fig. 5.1, is an oscillator in which the feedback path has a delay τd that is independent of frequency. Hence, when the amplifier gain A = 1, the Barkhausen condition is met at any frequency ωl = (2π/τd)l with l integer, that is, a frequency multiple of the free spectral range 2π/τd. With appropriate initial conditions, stationary oscillation takes place. If the exact condition A = 1 is met in a frequency range, several oscillation frequencies may coexist.

A real oscillator requires a small-signal gain A > 1 that reduces to unity in appropriate large-signal conditions. Of course, a selector filter is necessary in order to choose a mode m and thus a frequency ωm = (2π/τd)m. The filter introduces attenuation at all frequencies ωl with l ≠ m. For laboratory demonstration purposes, the imperfect gain flatness as a function of frequency of real amplifiers is sufficient to select a mode, albeit an arbitrary one. A true band-pass filter is necessary for practical applications. Using a tunable filter the oscillation frequency can be switched between modes, as in a synthesizer, in steps Δω = 2π/τd.

It will be shown that the laser is a special case of a delay-line oscillator. Additionally, the delay-line oscillator is of great interest for the generation of microwaves and THz waves from optics because a long delay can be implemented thanks to the high transparency of some materials, such as silica, CaF2, and MgF2, at 1.55 μm wavelength.

INTRODUCTION

We asserted in the previous chapter that tuned oscillators produce outputs with higher spectral purity than relaxation oscillators. One straightforward reason is simply that a high-Q resonator attenuates spectral components removed from the center frequency. As a consequence, distortion is suppressed, and the waveform of a well-designed tuned oscillator is typically sinusoidal to an excellent approximation.

In addition to suppressing distortion products, a resonator also attenuates spectral components contributed by sources such as the thermal noise associated with finite resonator Q, or by the active element(s) present in all oscillators. Because amplitude fluctuations are usually greatly attenuated as a result of the amplitude stabilization mechanisms present in every practical oscillator, phase noise generally dominates – at least at frequencies not far removed from the carrier. Thus, even though it is possible to design oscillators in which amplitude noise is significant, we focus primarily on phase noise here. We show later that a simple modification of the theory allows the accommodation of amplitude noise as well, permitting the accurate computation of output spectrum at frequencies well removed from the carrier.

Aside from aesthetics, the reason we care about phase noise is to minimize the problem of reciprocal mixing. If a superheterodyne receiver's local oscillator is completely noise-free, then two closely-spaced RF signals will simply translate downward in frequency together. However, the LO spectrum is not an impulse and so, to be realistic, we must evaluate the consequences of an impure LO spectrum.

A coherent image-formation system is degraded if the coherence of the received waveform is imperfect. This reduction in coherence is due to an anomalous phase angle in the received signal, which is referred to as phase error. Phase errors can be in either the time domain or the frequency domain, and they may be described by either a deterministic model or a random model. Random phase errors in the time domain arise because the phase varies randomly with time. Random phase errors in the frequency domain arise because the phase of the Fourier transform varies randomly with frequency. We will consider both unknown deterministic phase errors and random phase errors in both the time domain and the frequency domain.

Random phase errors in the time domain appear as complex time-varying exponentials multiplying the received complex baseband signal and are called phase noise. Phase noise may set a limit on the maximum waveform duration that can be processed coherently. Random phase errors in the frequency domain appear as complex exponentials multiplying the Fourier transform of the received complex baseband signal and are called phase distortion. Phase distortion may set a limit on the maximum bandwidth that a single signal can occupy. We shall primarily study phase noise in this chapter. Some of the lessons learned from studying phase noise can be used to understand phase distortion.