Intellectual Merit: The recent invention of f-2f self-referencing modelocked laser clockworks has heralded a new era in precision time and frequency metrology. In spite of phenomenal advances, there are still unanswered questions about the fundamental performance limits which might be expected from these clocks. This project will investigate an important component of laser clock performance; that of laser amplitude and phase fluctuations induced by noise in the pump source. The coupling mechanisms to be studied include perturbations to the: 1) complex susceptibility through the energy level populations, 2) length of the gain medium (thermal), 3) index of refraction (direct thermal), 4) index due to thermally-induced stress, 5) photon number, and 6) nonlinear index (Kerr-effect). These mechanisms combine to degrade the performance of the laser when the pump contains random noise. By inducing small-signal modulations on the pump beam, the magnitude of these effects will be measured and compared with theory. They will then be used to predict the ultimate performance of modelocked laser clocks and provide guidance toward better clock designs. Finally, f-2f interferometer control loops will be installed on two separate lasers to compare the closed-loop performance against predictions from theory.
Broader Impacts: Precision time and frequency standards form the very foundation of navigation, commerce, telecommunications and many branches of science and technology. Improved clock precision will affect all aspects of human life. It will be especially significant in pushing the frontiers of fundamental science as we explore atoms and molecules with ever-higher precision, test whether or not the ``constants of nature'' are truly constant and improve our theories of cosmology. The results of this work will be incorporated into classroom lectures in the Optical Science and Engineering program at UC Davis and will also attract the attention of students outside of the program who are naturally fascinated by the fundamental limits of technology and how they advance basic science.
All clocks or oscillators are accompanied by noise. Thus, there is no such thing as a perfect clock and the noise that accompanies all clocks degrades their stability. Viewed a different way, amplitude and timing fluctuations (jitter or phase noise) are the manifestations of an imperfect clock. Understanding the mechanisms that give rise to these noise sources and suppressing them results in improved clock performance. The goal of the NSF-funded research described in this report was to study mechanisms that contribute noise, and thus uncertainty, to the new class of laser-based clockworks. The mechanism studied was the imprinting of pump laser fluctuations onto the laser clock's output. This is shown schematically in Image 1. A laser with a large number of synchronized cavity modes (modelocked) forms a train of repetitive pulses. The period of this pulse train is usually a few nanoseconds and detecting the light pulses with a photodetector produces a stream of current pulses which are the heart of the clock. If the pump energy source has amplitude fluctuations (noise), the noise will transfer to the amplitude of the pulsed laser train as well as the timing between pulses. Thus, we may distinguish two types of noise in this pulsed laser oscillator; amplitude-modulation (AM) and phase-modulation (PM) noise. The fundamental mechanisms that link pump fluctuations with AM and PM noise of the clock laser are associated with three physical processes. Changes in pump power change 1) the number of photons inside the laser cavity, 2) the atomic population inversion, and 3) the temperature of the gain medium. The first two processes are intimately linked by rate equations and the third process in amenable to conventional heat transfer theory. Amplitude noise reflects the number of photons in the cavity and thus the rate equations tell us that pumping fluctuations cause synchronous cavity photon fluctuations until a frequency is reached at which the atomic population inversion can no longer keep up. This is called the "relaxation oscillation" frequency and is well known to laser scientists. The PM produced by pump noise is considerably more complicated and involves all three processes. Variations in all of these result in variations in the round trip time that photons inside the cavity experience and this means variation in the time period between clock pulses exiting the laser. For example, the number of photons inside the laser cavity will change the optical index of refraction (speed of light) inside any material such as the gain crystal due to nonlinear effects. This will, in turn, change the beam direction and round trip time by minute amounts for any beam refracting at other than normal incidence. When the population inversion changes, there is a direct change in the linear index of refraction and thus the propagation time through the gain medium. As the temperature of the gain crystal changes with pump power, the amount of heat deposited also changes which varies the physical length of the crystal as well as its index of refraction. There are additional higher-order effects that also contribute but are of much smaller magnitude. The characterization of the susceptibility of a laser to pump fluctuations should depend on the frequency of the fluctuation and be normalized to the magnitude of that fluctuation. This allows us to cast the laser performance in terms of a "transfer function"; a process that transfers pump noise to laser amplitude and timing noise. We have therefore cast our theory in terms of a laser "Noise Transfer Function" (NTF). The AM-NTF is defined as the fractional laser amplitude fluctuation per fractional pump power fluctuation at a specific modulation frequency. The PM-NTF is defined as the fractional change in pulse period per fractional pump power fluctuation. Image 2 shows examples of typical pump noise power spectra (a) and the magnitudes of the AM-NTF (H_AM) and the PM-NTF (H_PM) in (b). The AM-NTF shows us that a 1% pump fluctuation will cause a 2% power fluctuation below 500 kHz (relaxation oscillation frequency). However, in the flat portion of the PM-NTF, a 1% pump fluctuation will cause a 0.5 picosecond change in the 10 nanosecond period of a 100 MHz laser clock, or equivalently 50 parts-per-million. The key feature of our NTF characterization is the ability to use the NTF to predict the amplitude and phase noise of a laser clock from the noise spectrum of the pump source (Image 3). The NTF can be measured directly by using a modulator between the pump and the clock laser, or by calculating the AM-NTF and PM-NTF from first principles using the theory developed in this research program. We have done both and found very good correlation between them. We believe that this will lead to a better understanding of the fundamental limitations of laser clock performance and subsequent improvements upon the current state-of-the-art.
