Equivalence between free-electron-laser oscillators and actively-mode-locked lasers: Detailed studies of temporal, spatiotemporal, and spectrotemporal dynamics

We show experimentally and numerically that free-electron-laser (FEL) oscillators behave in a very similar way to conventional actively-mode-locked lasers. This stems from the similar structures of their underlying Haus equations. A comparative study of the temporal evolutions of the pulse train shapes and spatiotemporal regimes is performed on a Nd:YVO${}_4$ laser and a storage-ring free-electron laser. Furthermore, since direct observations of time-resolved pulse shapes and spectra are more accessible on free-electron lasers, the analogy also potentially enables one to investigate mode-locked laser dynamics using existing FEL facilities.

Figure 1

Paradigm of (a) free-electron-laser oscillators, (b) classical mode-locked lasers with a synchronous pump, and (c) classical mode-locked lasers with loss modulation. A main comparison point concerns the modulation of gain or losses at (or near) the round-trip frequency (or a multiple). HR denotes the high-reflection mirror and OC denotes the output coupler.

Figure 2

Illustration of the temporal evolution of laser losses, gain, and intensity for (a) and (b) FEL oscillators (and, more generally, synchronously pumped lasers) and (c) and (d) loss-modulated mode-locked lasers.

Figure 3

(a) FEL detuning curve calculated from Eqs. (1), (6), and (7). For each value of $v$, the maxima and minima of the pulse energy $I={\int }_0^L{\left|e(T,\theta )\right|}^{2}d\theta$ are represented. Also shown is the intensity of the pulse train envelope for (b) $v=0$ and (c) $v=4.7$. The parameters are $A=2$, ${\sigma }_b=846$, and $\eta ={10}^{-12}$.

Figure 4

Numerical results for the case of an actively-mode-locked laser. (a) Representation of the laser intensity as a function of the detuning parameter. Also shown is the laser intensity of the pulse train shape for (b) $v=0$ and (c) $v=1.15$. The normalized parameters used for the simulations are $R=1.2$, $\gamma =5\times {10}^{-4}$, $\mu =5\times {10}^{-8}$, $D=0$, and $\eta ={10}^{-14}$.

Figure 5

Experimental setup for the actively-mode-locked laser. A Nd:YVO${}_4$ laser crystal (5 mm long) that is high-reflection coated at 1.06 $\mu$m on the pump side ($M$1) and Brewster cut on the other side is shown. DL denotes the diode laser (Thales TH-C1610-F2), 10 W at 808 nm, fiber coupled (with a diameter of 200 $\mu$m and a numerical aperture of 0.22); $L$ denotes the aspheric lens; $M$2 and $M$3 denote high-reflection mirrors at 1.06 $\mu$m (with a radius of curvature of 50 and 20 cm, respectively); ML denotes the acousto-optic mode locker (IntraAction ML-503D1) with a Brewster-cut crystal; OC denotes the output coupler with a transmission of 5$\%$ and an antireflection face with a 3${}^{\circ }$ wedge, mounted on a motorized translation stage.

Figure 6

(a) Experimental detuning curve versus cavity length in the actively-mode-locked laser. The laser power is measured with photodiode resolving the envelope variation, but not the individual pulses. (b) and (c) Time traces recorded with a detector resolving the individual pulses (the repetition rate is 100 MHz): (b) mode locking with a stationary envelope observed near perfect tuning ($v\approx 0$) and (c) mode locking with a pulsed envelope behavior observed at finite detuning [the mirror has moved approximately $0.05$ mm from the (b) situation]. The experiment is performed near laser threshold ($R=1.2$).

Figure 7

Departure from the analogy between the free-electron laser and the mode-locked laser when the pump power is far above threshold. Detuning curves are recorded as in Fig. 6, versus pump power. The normalized pump power is (a) 1, (b) 1.1, (c) 1.3, (d) 1.7, and (e) 3.5.

Figure 8

Numerical results in the case of an actively-mode-locked laser. (a)–(c) Representation of the pulse intensity distribution (vertical scale) as a function of time (horizontal scale). (d)–(f) Representation of the pulse spectral distribution (vertical scale) as a function of time (horizontal scale). The normalized parameters used for the simulations are $R=1.2$, $\gamma =5\times {10}^{-4}$, $\mu =5\times {10}^{-8}$, $D=0$, and $\eta ={10}^{-14}$ for different detuning values: (a) and (d) $v=0$, (b) and (e) $v=0.15$, and (c) and (f) $v=8.8$.