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The amplitude and envelope phase noise of a modelocked Ti:sapphire laser are predicted based on the power spectral density of the pump laser and the noise transfer process. Pump laser noise is found to transfer directly to the modelocked laser’s amplitude and phase noise power spectra through the noise transfer function (NTF) which is independently measured. We find good agreement between the shapes and absolute values of the predicted and measured Ti:sapphire AM and PM noise spectra except in regions where additional environmental influences affect the Ti:sapphire laser. The experiments were conducted with both a single-mode and a multi-mode diode-pumped solid-state pump laser.
©2008 Optical Society of America
1. Introduction
The stability of modelocked lasers has taken on tremendous importance in recent years, especially since the invention of the carrier-envelope offset (CEO)-stabilized, self-referencing technique which forms the basis of the new laser clockworks [1, 2]. In spite of the impressive results obtained with this new technique, the performance of feedback control systems depends, at least in part, on the performance of the object being controlled prior to closing the loop. Thus it is still important to study, quantify, and thereby predict the noise properties of the free-running modelocked laser. This may lend insight into better laser designs and what mechanisms determine the ultimate stability of laser clocks.
For CW lasers, the issue of stability was initially one of linewidth which was shown to be limited by spontaneous emission [3]. In 1975, Haken [4] realized that the set of coupled equations describing the electric field, atomic polarization and population inversion were isomorphic to the equations derived by Lorenz to study Rayleigh-Bénard convection flow [5]. This gave rise to the explosive growth of the study of large-scale laser instability and chaos [6, 7]. Although these phenomena provided an interesting arena for experimental studies of a particular system of equations (closely related to weather), they are highly undesirable for most applications. Noise and “small-scale” laser instability form a different class of phenomena and are very relevant to modern technology, impacting everything from telecommunications and instrumentation to precision clocks and fundamental science.
Today, the forefront of ultrafast laser performance is dominated by passive modelocking and, in particular, the fast saturable absorber-like processes such as Kerr-lens modelocking. The analytical approach to understanding and designing these types of lasers that is widely used today is based on the “master equation” of Haus [8] and was further developed to describe colliding pulse (CPM) [9], additive pulse (APM) and Kerr-lens (KLM) modelocking [10]. The issue of applying the master equation to questions of noise and the stability of the solutions was addressed early on by Haus and Mecozzi [10] and more recently by Kapitula, et al. [11] and Menyuk, et al. [12]. What is common to most of these analyses is the goal of describing from first principles the mechanisms responsible for the intrinsic noise of the modelocked laser and the stability of the solutions when interrupted by a perturbation. A key finding from this work, and one that makes sense intuitively, is that fluctuations in the laser gain will have a direct impact on several of the measurable parameters of the modelocked laser.
With the advent of very low-noise diode-pumped solid-state (DPSS) pump lasers, which have become the norm for pumping KLM lasers, it seems reasonable that the modelocked laser might be simply treated as a linear system to which we can ascribe a transfer function that converts the noise of the pump to the important properties of the modelocked laser such as amplitude noise and timing jitter. These properties are, arguably, the most technically relevant and straightforward to measure.
The purpose of the study presented here is to demonstrate that there is a direct correlation between the noise properties of a modelocked laser’s pump source and the noise of the laser itself. Apart from environmentally-induced noise due to temperature fluctuations, air currents, acoustic vibrations, etc., and spontaneous emission noise, fluctuations in the pumping rate of any laser should be directly transferred to operating properties of the laser such as amplitude and timing stability, and therefore show up as additional noise. We can quantify the sensitivity of any laser to the fluctuations in its pump source by intentionally placing a small amount of modulation on the pump (stimulus) and measuring the effects on the laser (response) as we sweep the pump modulation frequency over the range of interest. By keeping track of both the amplitude and phase of the stimulus and response in this process we establish a complex noise transfer function (NTF) for the laser [13]. This characterization can be applied to both amplitude noise and envelope phase noise (timing jitter) of a modelocked laser. Once the NTF is known, we can predict the amplitude and phase noise power spectral densities (PSDs) of the modelocked laser by considering the NTF as representing the transfer function of a linear system. For a random process passing through any linear system, the output power spectral density is Sout(ω)=|H(ω)|2 Sin(ω) where H(ω) is the transfer function of the linear system and Sin(ω) is the input power spectrum. Thus, we can predict the laser output noise power spectra (either AM or PM) once we have the corresponding noise transfer function characterized.
The process of transferring pump noise to the AM and PM noise of a modelocked laser is shown schematically in Fig. 1. The baseband pump noise PSD, SP(ωm), is first measured over a wide range of frequencies (0.1 Hz–10 MHz). Next, the complex AM and PM noise transfer functions (NTFs) are measured over the same range. The NTF is defined as the ratio of the induced amplitude, m̃L(ωm), or phase, (ωm), modulation index to the pump modulation index, mp, at a given driving frequency, ωm [13],
The output amplitude noise PSD can now be predicted by taking the product of the pump noise PSD and the squared-modulus of the AM NTF
The single-sideband phase noise PSD is similarly given by
where the factor of 1/2 is due to the noise being specified as single-sideband about the carrier.
2. Measurement of laser noise power spectral density
The apparatus for measuring both AM and PM PSD is shown in Fig. 2. Under test are a diode-pumped solid-state pump laser and a conventional Kerr-lens modelocked Ti:sapphire laser with intracavity prism-pair dispersion compensation running at a 100 MHz repetition rate. The acousto-optic modulator (AOM) in the pump beam path is used for characterizing the NTF of the Ti:sapphire laser and is disabled during noise tests. Photoreceivers AM1 and AM2 are conventional transimpedance amplifiers used for measuring amplitude noise at baseband and are detailed in [14]. The envelope phase noise of the Ti:sapphire laser is measured at the fundamental of the 100 MHz pulse train using the quadrature mixing technique [14] with a voltage-controlled crystal oscillator (VCXO) as the reference. Photoreceiver PM2 contains a silicon p-i-n diode followed by a broadband RF amplifier. The spectrum analyzer system is composed of an HP 3561A (FFT) for the frequency range 0.1 Hz–100 kHz and an HP 3585A (analog RF) for the frequency range 100 kHz–10 MHz.
Fig. 2. Simplified block diagram of the setup used to measure laser noise and the complex noise transfer function. VSA, Agilent 89410 Vector Signal Analyzer; AOM, acousto-optic modulator; PD, photodiode; PLL, phase-locked loop; VCXO, voltage-controlled crystal oscillator; AM1 and AM2, baseband AM noise photoreceivers (DC-40 MHz); PM1, phase noise RF photoreceiver (80 MHz-120 MHz); LNA, low noise amplifier; Spectrum Analyzer, HP 3561A+HP 3585A.
For this experiment we chose to compare the performance of the modelocked Ti:sapphire laser using two different pump lasers: a single-longitudinal mode laser (Coherent Verdi V5 hereafter designated as Pump 1) and a multi-longitudinal mode laser (Spectra-Physics Millenia Vs, Pump 2). Although both are commercial frequency-doubled Nd:YVO 4 lasers, their noise spectra differ substantially as shown in Fig. 3(a). The additional noise seen in Pump 2 is attributed to mode competition and beating effects in the multi-longitudinal mode laser. (Note: a comparison between similar pump lasers was recently reported by Matos, et al. [15]).
3. Measurement of the laser noise transfer function
Like any transfer function, measurement of the laser noise transfer function requires a stimulus and measurement of a coherent response. This is accomplished using a vector signal analyzer (VSA, Agilent 89410A) which covers the frequency range 0.1 Hz–10 MHz. The VSA provides a drive signal that is applied to the AOM placed between the pump and Ti:sapphire laser. The VSA detects the amplitude and phase modulations induced on the Ti:sapphire laser (Ch. 2) as well as those on a sample of the pump beam (Ch. 1). The ratio of these establishes the NTF according to (1). Further details of the NTF measurement procedure can be found in [13].
The magnitude of the complex NTFs of the Ti:sapphire laser are shown in Fig. 3(b). Since we are only concerned with the absolute values of the NTF for the purpose of predicting the resulting AM and PM Ti:sapphire noise spectra, we omit the phase of this otherwise complex quantity. (Note: the PM NTF is only recorded down to 1 Hz due to the PLL loop filter [13]). Using both pump lasers to make the NTF measurements, we found the resulting noise transfer functions to be nearly indistinguishable. This is not surprising since the measurement of the transfer function of a linear network normalizes out the spectral shape of the source. In previous work, we also found that our measurements of the NTF did not depend on the modulation index applied to the pump [16]. These facts lend further credence to the hypothesis that for small pump fluctuations, the KLM laser truly acts as a linear system.
Fig. 3. (a) Pump laser noise PSD for a single-longitudinal mode DPSS laser (Pump 1) and a multi-longitudinal mode DPSS laser (Pump 2). (b) Magnitudes of AM and PM NTFs of the Ti:sapphire laser using Pump 1 and Pump 2.
4. Results
Based on the measurements of pump noise and the NTFs, AM and PM power spectral densities of the modelocked laser were predicted and compared to the measured PSDs. Figure 4 shows these predicted AM (a) and PM (b) PSDs along with the measured PSDs for the modelocked laser when it is pumped by Pump 1. The predicted PSDs were calculated by multiplying the measured pump noise spectra shown in Fig. 3(a) by the square of the respective NTF curves shown in Fig. 3(b). The noise measurements were made as discussed in Section 2. There are several interesting features to note. In the frequency range from ≈100-1000 Hz, the excess pump noise (due to line spurs and environmental effects) is transferred to both the AM and PM laser noise spectra. Also transferred are the large spurs at multiples of 100 kHz due to the pump laser’s switching power supply (see also Fig. 3(a)). In general, the shape and magnitude of of the measured PSDs track the predicted curves well except in the region of 50–1000 Hz. Here we believe that environmental effects such as air currents and acoustic vibrations affect the Ti:sapphire laser independently of the pump since it is an open frame system with no special provisions for acoustic isolation. We note that the impact is much more serious for the PM noise than for AM since it is the path length that is being perturbed rather than a gain-determining element. At very low frequencies (<2 Hz), we see that the measured noise rises above that which is predicted. This may be related to long term pointing instability in the pump laser or temperature fluctuations in the laboratory. Also, in the range 2–40 Hz the measured noise is less than that which is predicted. The pump laser has built-in servos which tune the diode arrays for maximum absorption in the Nd:YVO4 crystal almost continuously. The time constants and general behavior of the servo loops are unknown, but we observe that the overall low frequency noise behavior of the pump tends to fluctuate on a several second time scale and these phenomena may be related and the source of inconsistencies. Making simultaneous pump and modelocked laser noise measurements should reduce these effects.
Fig. 4. Measured and predicted (a) AM and (b) PM noise of a KLM Ti:sapphire laser pumped with a single-mode DPSS laser (Pump 1). (Note the different scales).
Fig. 5. Measured and predicted (a) AM and (b) PM noise of a KLM Ti:sapphire laser pumped with a multi-mode DPSS laser (Pump 2). (Note the different scales).
Figure 5 shows the predicted and measured AM (a) and PM (b) PSDs for the Ti:sapphire laser pumped by Pump 2. The previous comments for Pump 1 results apply here as well but the pump laser has higher overall noise which is reflected in the Ti:sapphire noise. Also, there are fewer spurs in the pump spectrum from 100 kHz upward and this is presumably due to a different power supply configuration. For both pump lasers, above the relaxation oscillation frequency (ROF) both the AM and PM NTFs roll off at -20 dB/dec. Applying this rolloff to flat pump noise predicts a -20 dB/dec slope in the PSDs of the Ti:sapphire at high frequencies. This rolloff does not occur in the measured PSDs however due to the presence of shot noise. Also, the presence of spurs that occur only in either the predicted or measured PSDs but not both is due to intermittent environmental noise.
5. Conclusions
The AM and PM noise spectra of a free-running modelocked laser appear to be dominated by the influence of pump noise and can therefore be predicted by characterization and application of the noise transfer functions to the AM noise spectrum of the pump. We applied this procedure to a single- and multi-longitudinal mode DPSS pump laser. The predicted AM and PM noise spectra are in reasonably good agreement with the measured spectra for both pumps.
Acknowledgment
This research was supported in part by NSF Grant ECS-0622235 and the David and Lucile Packard Foundation. The authors thank Bob Temple of Agilent Technologies for many insightful discussions and technical support.
References and links
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