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Self-organization of the dynamic cavity completed by holographic gratings in the novel laser oscillator was studied experimentally and numerically. A key role of the resonant grating of refractive index induced by generating beams in an Nd:YAG laser crystal was determined. Stabilization of the generated pulses and an increase in output power were achieved by use of a vibrating intracavity mirror.
©1999 Optical Society of America
1. Introduction
The cavity of a laser oscillator can be formed by resonant dynamic gratings induced in an active medium by the interference field of generating waves. The observation of self-starting generation in laser oscillators of such kind has been reported previously for solid-state active media [1]. Currently, the self-starting solid-state lasers with the dynamic cavity are being intensively investigated [2–7]. The main advantages of these laser oscillators are self-adaptation and self-Q-switching of the dynamic cavity with nonlinear mirrors. These unique properties of the lasers of a new class provide a good beam quality, a high pointing stability, and a large coherence length of the generated radiation.
Two types of the self-starting laser-oscillators with dynamic cavity have been recently demonstrated. The first type, based on Nd:YAG and Ti:S laser crystals (LaCs), incorporated a nonreciprocal transmission element in the loop section of the cavity formed by the gain grating (GG) [2,3,6]. Another laser oscillator with reciprocal cavity was reported to be formed by moving resonant refractive index grating (RIG) accompanied the population grating (PG) induced in a Nd:YAG crystal by the generating beams [4,5]. The latter type of the laser has demonstrated the capability for generating beams with high average power (as large as 60W) and near-diffraction-limited quality [7]. In this article, we present the results of numerical and experimental investigations of the origin and dynamics of the gratings that complete the reciprocal cavity of a self-starting high-average-power Nd:YAG laser.
2. Principles of laser cavity formation by self-induced refractive index gratings
The general idea of the laser oscillator with a cavity completed by the holographic RIG induced in a LaC can be described for the simplest example of a laser scheme consisting of a Nd:YAG amplifier and a loop feedback formed with mirrors M1-M3 (Fig. 1).
Fig. 1. Schematic of a self-starting generator with a loop cavity. M1 is the vibrating mirror that gives the frequency shift Ω to reflected waves; k1,…k4 are the wave vectors of the interacting optical waves E1,…E4 in the cavity; the traits indicate the PGs inside the LaC.
Four optical waves (starting initially from amplified spontaneous emission (ASE)) with complex amplitudes E1 ,…E4 can interfere with each other. The interference fields will induce the PGs, whose complex amplitudes nij can be given by:
where i, j=1, 2…4 (i≠j); Nη is the amplitude of the Langevin noise sources for the PGs; ηij ,ψij are random delta-correlated functions normalized to 1; time and wave intensities are normalized to the longitudinal relaxation time of working transition and its saturation intensity, respectively; and N0 is the average-in-space population, which is given by:
where Np (t) is the pump velocity determined in experiment.
The gratings of population inversion in the laser crystals are accompanied by both GGs and RIGs. The latters are caused in the Nd:YAG amplifier by the difference in polarizability of excited and unexcited Nd3+ ions [8]. The RIGs and, generally, the resonant changes of refractive index can be taken into consideration by the real part of the nonlinear resonant susceptibility [7,8]. The interaction of the generating waves in LaC by the mutuial scattering of the waves both on the RIGs and the GGs is described in the one-spatio-dimensional approximation by the following set of equations (an analoguos set of equations was used for the steady state analysis of the conditions for the self-starting generation [7]):
where σ=σ0 (1+iβ), σ0 is the cross-section of the resonant transition; β is the ratio of the real part of the Nd:YAG resonant susceptibility to the imaginary part [8]; µ=l/c is the walk-off time in the rod; F ε is the amplitude of the Langevin noise source for the optical field; εi (z,t) and φi (z,t) are random delta-correlated functions; N12 =N34 =n12 +n34 , N13 =N24 =n13 +n24 , N14 =n14 , N23 =n23 , Nij = ; ‘+z’ corresponds to the propagation direction of the waves E1 and E3 .
The Langevin noise sources in Eqs. (1) and (3) determine the initial levels of the PGs and the optical waves, providing starting conditions of the laser oscillator. In practice, the population noise is produced by pump fluctuations and spontaneous resonant transitions and is induced by the broadband nonresonant ASE; the noise source of the optical field occurs as a result of spontaneous polarization and thermal optical field in the cavity [9].
In order that the system shown in Fig. 1 could operate as a laser, the induced gratings must provide positive feedback by energy transfer from the strongest wave E4 to the weakest wave E1 . Such energy transfer can be realized by the GGs in the presence of a phase-nonreciprocal element in the loop [2,3,6]. In the case of a reciprocal loop cavity only the moving RIGs that participate in two-wave and four-wave interactions inside the Nd:YAG amplifier can provide the energy transfer [4,5,7]. The movement of the grating leads to the compensation of the π/2-phase shift between the RIG and an additional amplification coefficient of the weak waves. In our scheme the moving RIGs are induced by the interference field of the optical waves with frequency detuning, in this case the phase matching of the waves in the cavity occurs as a result of reflection from the moving intracavity mirror [5,7].
The numerical solution of Eqs. (1)–(3) was made for zero initial condition, mirror boundary conditions (100% reflection from the mirrors M1-M3), and the weak diffusive reflection (rdif ) of the wave E4 at the output boundary of the Nd:YAG rod. It was calculated that the nonlinear generation occurs in the presence of the PG noise source (when Nη ≠0), at a very small value of the noise source for the optical wave that propagates in the direction of generation (F ε≈10-7÷10-8), and even without any “linear” diffusive reflection on the rod end rdif =0 (Fig. 2).
Fig 2. Results of numerical calculations: (a) oscillograms of the output wave I4 when maximum of the unsaturated amplification coefficient is α0l=1.5 (red curve), α0l=2.0 (blue curve), and α0l=4.0 (violet curve); the frequency detuning is Ω=1.25 (red and blue curves) and Ω=200 (violet curve); the PG noise is Nη =1.39×10-3; brown curve is the temporal profile of the Nd:YAG amplification; (b) oscillograms of the population grating N13 (blue curve) and the output wave intensity I4 (pink curve) when α0l=4.0 and Ω=250; (c) dependencies of the peak-pulse intensity I4 (solid curves) and the energy W (dashed curves) of the output wave on the amplitude of the PG noise source at Ω=1.25 (red curves), Ω=1.5 (blue curves), and Ω=1.0 (green curves), when α0l=2.0, F ε=10-8.
The numerical calculations showed that at a fixed level of the noise source amplitudes the generation had the threshold for the unsaturated amplifier gain (α0 =σ0 N0 ) that depended on the frequency detuning (Ω) of the optical waves inside the cavity (Fig. 2a). Below the threshold for the amplifier gain only noise amplification took place (red curve in Fig. 2a), at the threshold the single-pulse generation was realised (blue curve), and beyond the threshold the pulse train generated (violet curve).
The investigation of PG indicated that the generation started when the RIG diffraction efficiency achieved a threshold level that is determined by the amplification and losses in the cavity. The self-starting generation was not observed in the absence of the RIG (β=0). At the beginning of the first peak of generation (and several subsequent peaks) the PG amplitude and the optical wave intensity increased self-consistently (Fig. 2b). The PG growth continued till the gain saturation by the high-intensity optical waves.
At the fixed amplifier gain the generation intensity strongly depended on the amplitude of the PG noise source Nη (Fig. 2c). The minimum of generation threshold for Nη corresponded to the frequency detuning Ω=1.25. The generation Ω-band expanded at the large gain. For example, at α0l=4 the difference in the generated energy for Ω=1.25 and 250 was about 10%.
3. Experimental results
The cavity of the laser oscillator we studied experimentally was formed by ordinary mirrors, a Sagnac interferometer (which provided spatio-temporal mode selection [7]), and holographic gratings induced inside the Nd:YAG amplifiers (Fig. 3). The laser amplifiers (based on Nd:YAG crystals with 1% concentration of Nd3+ ions) had logarithmic gain up to 4.8 at a flash-lamp-pump pulse duration of 0.3 ms and a pump-pulse energy as large as 80 J per each rod.
Fig. 3. Schematic view of the NDFWM investigation of RIG formation in a laser with a dynamic loop cavity. Diagram of the wave vectors for NDFWM inside Nd:YAG rod is presented separately.
The self-starting generation was observed in this scheme when the total amplifier gain exceeded a threshold level. This generation cannot be explained by any linear scattering of the output beam because it was directed in a black body with negligible backscattering.
To study the origin and the dynamics of the holographic gratings completing the cavity the nondegenerate four-wave mixing (NDFWM) measurements were made using an additional Nd:phosphate glass laser (Fig. 3). The intersecting generation waves E2 and E4 (at the resonant wavelength of 1064 nm) induce the PG in the Nd:YAG amplifier. The large-scale RIG that accompanies the PG was read by an optical wave of the Q-switched Nd:phosphate glass laser at a wavelength of 1054 nm with a repetition rate less than 1 Hz. The propagation direction of the reading beam (ER ) was chosen as optimal for Bragg diffraction on the recorded grating and was nearly opposite to the direction of one writing generation beam so that wave synchronism took place. The energy of the testing single-transverse-mode beam (20 mJ) was much less than thresholds of self-induced nonlinear optical effects. The diffracted beam (E D), which occurred as a result of reflection of the reading beam from the RIG, was recorded. This recorded signal was really diffracted on the RIG since it was observed only in the presence of the writing beams of generation and of the reading beam, simultaneously. The use of orthogonal polarisations of the reading and generation beams, as well as the different pulse durations of the beams and the pump, made negligible the level of the noise caused by the writing beams or spontaneous emission in the recording channel of the diffracted beam. Therefore, the NDFWM experiments confirmed the existence of the RIG inside the amplifier.
The pulse of the testing laser beam with a duration of 160 ns was synchronised with some variable delay at the beginning of the generation pulse. By changing this delay time, we studied the temporal dynamics of the tested RIG. The measured diffraction efficiency (DE) of the RIG, defined as the ratio of energies of the diffracted and reading pulses, depended on the delay of the reading pulse at the beginning of generation. It was measured that a noise signal was approximately the same in the nonpumped Nd:YAG and in the pumped LaC before generation. The DE strongly increased (by more than one order of magnitude) after the beginning of laser generation and decreased after its end (Fig. 4).
Fig. 4. Dependencies of the diffraction efficiency of the RIG on time for time scales of 5 µs (a) and 100 µs (b): experimental data for DE (▪), polynomial fit for DE (red line), typical oscillogram of the generated pulse (blue line), oscillogram of the single-pass unsaturated amplification coefficient of one amplifier (violet line), and oscillogram of the flash-lamp pumping (green line).
Measurements on the fast time scale showed that the DE of the resonant RIG began to increase several microseconds before the onset of generation (Fig. 4a). Formation of the nonlinear RIG mirror that completes the cavity preceded the first optical generation spike.
The DE of the RIG at the time of generation onset was approximately 5·10-7. Knowing this value of the nonlinear mirror “reflectivity”, we can estimate the threshold gain of the amplifiers in the self-starting generator. For generation to start in the cavity completed by the RIG, the total amplification should be equal to the total losses, which are caused mostly by the RIG transmission. Then, the single-pass amplification (K) of each Nd:YAG amplifier at generation threshold must be Kth ≈(5·10-7)-0.25=44.7. This estimation of Kth is in good accord with experimental measurements of the gain of the Nd:YAG amplifier at the generation threshold (Fig. 4b). Therefore it is possible to conclude that the generation in the studied scheme was indeed caused by the self-consistent processes of the RIG formation and by the increase in the amplitude of the generated waves that were reflected from the RIG.
The generation pulses and the DE of the RIG were recorded in the presence of some frequency shift between waves that write the RIG. This shift was realized by means of the mirror M1 placed on a piezoelectric vibrator. When an ac voltage with the piezoelectric-resonant frequency of 80 kHz was applied to the piezoelectric element, the synchronisation of generated pulses with a definite phase (φ) of the oscillating mirror and an increase in the pulse amplitudes were observed (compare Fig. 5a and 5b). This fact can be explained by that this phase of vibration corresponds to the optimal Doppler frequency shift of the waves reflected from the moving mirror. The presence of the frequency shift optimal for the laser generation in this experiment is in good accordance with the numerical calculations that defined the resonant frequency shift provided by the moving intracavity mirror. The observed generation in the absence of piezo vibration (Fig. 5a and 5e) can be explained by noise vibrations of all mirrors.
Fig. 5. Oscillograms of the laser radiation (top) and the voltage modulating the mirror displacement (bottom): piezoelectric vibrator switch off (a, e), piezoelectric vibrator switch on (b-d), the modulation frequency is ν=80 kHz. (b); ν=1.5 kHz, the phase φ is nonoptimal (c); ν=1.5 kHz, the phase j is near-optimal (d). The flash-lamp pump energy per each amplifier is 32 J for (a, b), and 50 J for (c)-(e). The vertical scales are the same for all oscillograms.
It was also observed experimentally that the duration of the generation pulses decreased from 400 ns to 200 ns in the presence of mirror vibrations. Such temporal behaviour of laser generation seems to be similar to the Q-switching in a laser with an ordinary mirror. However, in the studied laser with a nonlinear mirror the energy of the pulse train slightly increased because of the self-Q-switching (Fig. 6a), which is the difference from the ordinary Q-switch.
Another generation dynamics was realised when the period of the piezoelectric mirror oscillations was much greater than the duration of a generation pulse (Fig. 5c-5e). In contrast to the high-frequency modulation, neither pulse shortening nor ordering of the repetitive pulses was observed. On the other hand, the energy of the generation pulses was found to be considerably increased at the “true” phase of the mirror vibrations (Fig. 5d and 6).
Fig. 6. Dependencies of the energy of the generation pulse W and the DE of the RIG on (a) flash-lamp-pump energy per each amplifier and (b) phase φ of the voltage that modulates vibrations of an intracavity mirror. In Fig. 6a the mirror vibration is switched off (black curve), the mirror vibration is at a frequency of ν=80 kHz (red curve), and at 1.5 kHz with optimal phase (blue curve). Fig. 6b shows the dependence of the DE (squares and fitted green curve) and the dependence of the energy of generation pulse (circles and fitted braun curve); the horizontal lines are the energy (top) and the DE (bottom) when the modulation was switched off, and the flash-lamp pump energy is 50 J per each amplifier, ν=1.5 kHz.
These dependencies show that the mirror vibrations at a frequency of 1.5 kHz with an optimal phase led to a strong growth of the RIG diffraction efficiency, a decrease in the generation threshold, and to an increase in the peak pulse power and the pulse train energy. These results can also be explained by the resonance of the frequency detuning of the generating waves which induce the moving RIGs that complete the laser cavity.
Therefore, the experimental consequence is that the use of the moving mirror inside the cavity of the self-starting laser allows us to control the dynamics of the generated beam and increase the DE of the RIG, which results in an increase in the generated pulse-train energy. Note that the growth of the RIG by use of the vibrating mirror improved the spatio-temporal quality of the generated beam (which was demonstrated in Fig. 5). However, in the presence of strong vibrations of the scheme components (which appeared, in particular, because of the amplifier pump with a repetition rate of 10–30 Hz) the influence of the mirror movement on the generation characteristics decreased.
4. Conclusion
The theoretical and experimental investigations have shown that the reciprocal cavity of a self-starting Nd:YAG laser oscillator can be completed by resonant RIGs, which are induced in the inverted laser crystal by generating beams. The self-consistent increase of the moving RIGs and the optical waves beyond threshold results in the generation of self-Q-switched pulses. The self-starting Nd:YAG laser with cavity on the RIG is capable to generate high-average-power beams with good quality. Stabilization of the generating pulses and an increase in the output power are achieved by use of a vibrating intracavity mirror. The control of the vibrating mirror offers the possibility of changing parameters of the generating waves.
5. Acknowledgments
This work was supported in part by the EOARD through grant No SPC99-4028 and by the INTAS through grant No I 97-2112.
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