1. Introduction

With the advent of self-referenced modelocked lasers providing phenomenal pulse-to-pulse timing stability [1], it is important to study the mechanisms that set fundamental limits on the performance of these oscillators. In addition to spontaneous emission [2] and shot noise, a critical aspect affecting the amplitude and phase noise performance of the modelocked laser is noise accompanying the pump source. This holds true for optical or electrical pumping and whether or not the laser is part of a feedback loop. In addition, a complete characterization of the laser’s response to pump power variations is essential for controlling carrier-envelope offset and general timing stabilization [3–5].

Pump power fluctuations lead to several effects which modify the amplitude and pulse-to-pulse timing stability of the pumped laser. For example, variations in the pumping rate cause variations in the cavity photon number and thus the output power for slow variations. Above the relaxation oscillation frequency (ROF), this process is impeded as the population levels cannot keep up with the changing pumping rate. We call the complex, frequency-dependent sensitivity of the output power to pump fluctuations the AM noise modulation transfer function or AM NTF [6,7]. We reserve the designation MTF for applications involving spatial frequencies.

To characterize the AM NTF, we apply a small sinusoidal modulation with index m_P on the pump laser and measure the induced modulation index m̃_L on the output of the mode-locked laser. The ratio of these quantities at the modulation frequency ω_m defines the AM NTF: H_AM(ω_m) ≡ m̃_L(ω_m)/m_P.

Pump power fluctuations also produce envelope timing instability (phase noise) and we call this sensitivity the PM noise transfer function H_PM(ω_m). There are several mechanisms that couple the pump AM to laser PM [6,7]. These include temperature-dependent index of refraction, longitudinal thermal expansion of the gain medium, stress-induced change in the index of refraction, nonlinear index of refraction n ₂, and beam steering in the gain medium [8]. Additionally, the changing population levels modify phase shift through the complex susceptibility (which is only clamped in true steady state).

For a small-signal sinusoidal pump modulation, m_P, we can write the output power of the laser as a series of pulses with shape A(t),

where τ_co is the unperturbed round-trip cavity time, β̃(ω_m) is the phase modulation index (also called peak phase deviation) of the fundamental component of the Fourier series expansion of P(t), and ϕ(ω_m) is a frequency-dependent phase offset. In the frequency domain this waveform will exhibit PM sidebands in the detected photocurrent and from the magnitudes of these sidebands we can deduce the phase modulation index, β̃(ω_m) [9]. The ratio of this index to that of the pump defines the PM NTF: H_PM(ω_m) ≡β̃(ω_m)/m_P.

Previously, we reported the magnitudes of the NTF [6,7] but since H_AM(ω_m) and H_PM(ω_m) are complex functions of frequency it is essential to further characterize the complex behavior so that realistic estimates of the ultimate timing stability may be made when contemplating a closed loop system. In this paper we present results of measurements which characterize the complete complex sensitivity of a modelocked laser to fluctuations in the pump power across a wide range of modulation frequencies (0.1 Hz < f_m < 10 MHz). We find that the data are in good agreement with a linearized model of the laser based on the coupled cavity photon rate equations [10] combined with the solution of the heat diffusion equation.

2. Experimental setup

 

Fig. 1. Simplified block diagram of the setup used to measure 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 photoreceivers; PM1, high-frequency photoreceiver.

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Our approach to characterizing the NTF is similar to early work studying relaxation oscillations and spiking behavior [11] except that we use a coherent detection system. This allows us to record both the magnitude and the phase of the induced modulations. Figure 1 shows a simplified schematic diagram of our measurement setup. The laser under test is a Kerr-lens-modelocked (KLM) Ti:sapphire laser (KMLabs, model TS) with prism-pair dispersion compensation operating at a 100 MHz repetition rate. Typically, it has 20 fs output pulses and 0.4 W of output power with 5 W pump power (Coherent, V5 diode-pumped solid state (DPSS) laser). An Agilent 89410 Vector Signal Analyzer (VSA) supplies a drive signal at frequency f_m to an acousto-optic modulator (AOM) between the pump laser and the modelocked Ti:sapphire laser. This produces a weak (< 0.1%) amplitude modulation on the pump. A small amount of pump power is detected by photoreceiver AM1 to provide a reference signal to channel (Ch) 1 of the VSA which is proportional to the pump modulation index m_P. Direct measurement of the actual pump modulation obviates the need to correct for the AOM’s response. To measure the induced modulation on the Ti:sapphire output, separate receivers are used for the amplitude and phase measurements. Each receiver is optimized for its purpose and specific frequency region [9]. The AM receiver consists of a relatively large-area photodiode and a low-noise transimpedance amplifier with a -3 dB bandwidth of 1 Hz–60 MHz while the PM receiver is a photodiode whose output is lowpass filtered and terminated into the input of a low-noise RF amplifier (50 Ω) with a composite -3 dB bandwidth of 10–150 MHz. The modelocked laser’s amplitude modulation index m̃_L(ω_m) is obtained from photoreceiver AM2 driving Ch 2 of the VSA. After calibration, the ratio between the signals of Ch 2 and Ch 1 measured over the complete modulation frequency range gives the complex AM noise transfer function H_AM(ω_m).

Induced envelope phase fluctuations are measured using closed-loop quadrature phase detection with AM rejection ≥ 30 dB. A 100 MHz voltage-controlled crystal oscillator (VCXO) is phaselocked to the fundamental Fourier component of the modelocked laser’s pulse train. The output of the phase detector is sent to Ch 2 of the VSA and is directly proportional to β̃(ω_M). The ratio of this signal to the pump modulation index at Ch 1 gives the PM NTF H_PM(ω_m).

Calibration of the AM NTF measurement is accomplished in two steps. First, the average received photocurrents are equalized, assuring the ratio of the two modulation indices is equal to the ratio of their modulation sidebands, which are the signals that the VSA actually measures. Second, the complex frequency response of the two AM receiver signal paths are measured with the VSA and normalized out of the raw data. Calibration of the PM NTF is moderately more complicated since the PM sidebands are demodulated by the quadrature mixer before detection at the VSA. The frequency response of this operation is measured using the VSA and an Armstrong modulator which couples PM sidebands of known amplitude onto the detected 100 MHz carrier from PM1. Thus, the response of the PLL, mixer, and amplifiers are all taken into account and normalized out of the raw data. A spot measurement of both m_p and β̃ is then used to calibrate the magnitude of the PM NTF response.

3. Theory and results

For very small fluctuations in pumping rate, we may use the linearized coupled cavity rate equations to analyze the behavior of this otherwise highly nonlinear system (larger-scale nonlinear effects require application of the full nonlinear master-equation [12,13]). To validate the linearized model, we found that both H_AM(ω_m) and H_PM(ω_m) are invariant to changing m_P [6].

Small-signal fluctuations in pump power cause corresponding fluctuations in three physical quantities; photon number n(t), population inversion N(t) and laser rod temperature T(t). If we model the pumping rate by a strong steady-state component and a small-signal harmonic term

(R _P1/R _P0 ≡ m_P), this will result in sinusoidal variations in the physical quantities

where n_ss, N_th and T_o are the steady-state photon density, threshold population inversion density and steady-state rod temperature, respectively. Solving the coupled-cavity rate equations [10] and diffusion equation [14] with Eq. (2) as a source and Eqs. (3) as solutions, we find the phasor amplitudes of these quantities;

where ω_sp and γ_sp are the ROF and decay rate, respectively, γ_c is the cavity decay rate, k = 1.8×10^-5m²/s is the thermal diffusivity, r and b are the laser rod radial coordinate and radius, respectively and β_l are the roots of the thermal diffusion eigenvalue problem (a complete derivation of Eq. (6) will be presented elsewhere).

 

Fig. 2. Magnitude (a) and phase (b) of the measured and theoretical complex AM NTF response. Parameters: γ_c = 1.28 × 10⁷ s^-1, ω_sp = 2π × 700 kHz, γ_sp = 5×10⁶ s^-1.

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The AM-AM sensitivity, H_AM(ω_m), is caused by simple pump-induced fluctuations in the cavity photon density, ñ ₁. Figure 2 shows the theoretical fit (Eq. 4) and results from measurements of H_AM(ω_m) for modulation frequencies from 0.1 Hz to 10 MHz. The solution (Eq. 4) contains a second-order pole at the ROF. This is readily observed in the data which demonstrate the requisite -20 dB/dec rolloff above ω_m = ω_sp.

The origins of the AM-PM sensitivity, H_PM(ω_m), are considerably more involved [7,15]. At low modulation frequencies, thermal effects dominate and thus should follow the single pole behavior of Eq. (6). These include thermal expansion of the laser rod and the direct thermo-optic effect. At mid-range and higher frequencies, the high peak powers of the modelocked pulses cause variations in the group delay through the nonlinear index of refraction and thus follow the characteristics of the photon density Eq. (4). Modulation of the population inversion, Eq. (5), causes variation in the group delay in the gain medium because of the slope of the real part of the susceptibility, χ′(ω). This effect would show a combined double-pole, simple-zero at the ROF according to Eq. (5), but this is not observed indicating the weakness of the effect.

Figure 3 shows the measured data and the theoretical fit as magnitude (a) and phase (b) of H_PM(ω_m). The straight-line trends on the log-log plot of the magnitude show the single-pole (-10 dB/decade) behavior characteristic of thermal fluctuations [Eq. (6)] from 1 Hz to 300 Hz where the photon density fluctuation [Eq. (4)] takes over. Above the ROF (≈ 700 kHz), this response begins the characteristic -20 dB/decade rolloff of a double-pole filter. As described at the end of Sec. 2, the PM NTF measurements are taken outside of the PLL loop that holds the mixer in quadrature with the signal from PM1 (Fig. 1). Owing to the highpass-like PLL filter response and the relatively large close-in phase noise of the modelocked laser, both the magnitude and phase components of the PM NTF could not be reliably obtained below a modulation frequency of about 5 Hz. Instead, we employed a “spot” measurement technique using an HP 3048A phase noise system operating in real-time mode. This permitted magnitude-only data to be acquired from 0.1 Hz–10 MHz, thus filling in the vacancy in the first decade and corroborating the complex data taken with the VSA (Fig. 1). The spot measurement data are shown as yellow triangles in Fig. 3(a).

 

Fig. 3. Magnitude (a) and phase (b) of the measured and theoretical complex PM NTF response. Independent “spot” measurements of the magnitude are shown as triangles.

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4. Conclusions

In conclusion, we have measured the complex noise modulation transfer function of a Kerr-lens modelocked Ti:sapphire laser. The frequency dependence of the AM-AM and AM-PM coupling of pump laser modulation follows a simple linearized relaxation oscillation model combined with a solution to the heat diffusion problem. The complex noise transfer function and the technique used to measure it can be applied to characterizing all types of modelocked lasers. With this information, improvements in fundamental laser performance can be realized and then applied to closed-loop systems such as f-2f interferometers in laser-based clockworks.