Coupled-cavity passively Q-switched two-Stokes microchip laser

Abstract

Experimental and theoretical studies of the coupled-cavity diode-pumped Nd:YAG/Cr:YAG microchip lasers with intracavity Raman conversion of laser pulses in a Ba(NO₃)₂ crystal into two Stokes pulses have been made. Two lasers with a different cavity length have been investigated. The minimal pulse durations at the 2nd Stokes wavelength were ≈100 ps in the short-cavity laser at pulse energy of 5 μJ, and the pulse repetition rate reached 20–24 kHz. The laser and Stokes pulse dynamics, as well as the spatial intensity distribution of the laser and the 1st Stokes beams at the output mirror have been recorded. A model describing such coupled-cavity microchip Raman lasers has been developed. The numerically simulated laser and Stokes pulse dynamics, and the calculated pulse energy, duration, and repetition rate are in good agreement with the experimental data.

1 Introduction

In the last decade, diode-pumped microchip lasers have been generally recognized as compact, robust, and relatively cheap commercially fabricated devices having a large potential for applications as laser sources in scientific and industry fields [1, 2]. Peak pulse powers in microchip lasers achieve a few dozens of kW [1–8], which is enough to ensure efficient nonlinear Raman conversion. There are several works devoted to the Raman conversion of the radiation of pulsed microchip lasers with passive [9–14] and active [15–19] Q-switching. It has been demonstrated that high peak powers permit obtaining efficient generation of intracavity harmonics [20–23] and intracavity sum-mixing of the laser and Stokes radiation [24]. Also, via the process of Raman conversion, it is possible to obtain Stokes radiation with a higher peak power due to the well-known phenomenon of Raman compression [12]. In the last few years, diffusion-bonded composite Nd:YAG/Cr:YAG crystals have been widely used in microchip and mini lasers [6, 7, 19]. Their advantages are lower losses of laser and Stokes radiation due to the smaller number of optical surfaces and smaller surface bending. Recently the process of generation of first- and second-order Stokes pulses in an Nd:LSB microchip laser with a Cr:YAG passive absorber and a BaWO₄ Raman shifter has been investigated by us experimentally and theoretically [25]. In that case, Raman conversion took place in the same cavity as laser generation, the duration of 1st and 2nd Stokes pulses was 0.6 ns, and the energy of the 2nd Stokes pulses was less than 0.5 μJ. In the present work, we suggest to use, instead of a common cavity, a coupled cavity scheme for Raman conversion, in which a special intracavity mirror forms a short cavity for Stokes radiation. Such a scheme has several advantages. First, the intracavity mirror permits avoiding Stokes radiation absorption in the passive absorber. Second, due to the 1.5–3 times shorter cavity length Stokes pulses with shorter durations can be obtained. And, finally, to generate 2nd Stokes radiation one has to provide high reflectivity of the Raman cavity mirrors in the 1.064–1.4 μm wavelength range. For 2nd Stokes radiation pumped by 1.064 μm radiation, the wavelength lies in the 1.3–1.4 μm range where a strong laser line of Nd:YAG at the 1.34 μm wavelength exists. Spurious generation of 1.34 μm radiation in Nd:YAG can significantly lower the lasing efficiency at the 1.064 μm line and leading to a decrease in the Stokes generation efficiency. Thus, the use of an input mirror with a high reflectivity at 1.064 μm and a low reflectivity at the other wavelengths involved in combination with an intracavity mirror that is transparent at 1.064 μm and highly reflective in the 1.2–1.4 μm wavelength range is really advantageous for efficient generation of multiple Stokes radiation.

In this work, we present the results of the experimental and theoretical investigations of the Raman conversion at two Stokes wavelengths in Nd:YAG/Cr:YAG/Ba(NO₃)₂ microchip lasers with coupled laser and Raman cavities. Experimental data on the pulse energy, the pulse durations, and the pulse repetition rate have been obtained for different pumping levels. The pulse energy of the 2nd Stokes radiation at the 1.37 μm wavelength reached 5 μJ, and the pulse duration was as short as 100 ps, which corresponds to a peak power of 40 kW.

Section 2 describes in brief the experimental setup and the diagnostics of the temporal and spatial characteristics of the beams. Section 3 describes our numerical model for the microchip laser with Raman conversion in coupled cavities. This new model differs from the previously developed one [25] in two aspects. First, the equations for wave amplitudes were modified in order that they should comply with the scheme of coupled cavities. Second, the model takes into account the features of the laser generation of the Nd:YAG crystal at the 1.064 μm wavelength. Compared to the Nd:LSB laser crystal used in [25], stimulated emission of radiation at the 1.064 μm wavelength in the Nd:YAG crystal takes place in two channels corresponding to the radiative transitions of neodymium ions from the two laser levels in the upper ⁴F_3/2 multiplet down to the two laser levels in the lower ⁴I_11/2 multiplet [26]. These transitions have slightly differing wavelengths (1.0644 and 1.06415 μm), but their stimulated emission cross-sections (1.2×10⁻¹⁸ cm⁻² and 6.5×10⁻¹⁸ cm⁻²) differ widely [26, 27]. Thus, in pulsed lasers, these two channels have different dynamics of level replenishment and depletion and make a different contribution to the pulse development. The proposed model takes into account the transitions in both channels. The model takes into account the thermalization of neodymium ions within the laser multiplets ⁴F_3/2 and ⁴I_11/2 because the thermalization times are very close to pulse durations in microchip lasers. In the numerical calculations, we use the real spatial distributions of laser and Stokes intensities at the output mirror because they are almost unpredictable in microchip lasers with unstable plane cavities and strongly depend on the pump power.

Section 4 describes the results of the experimental investigation and the numerical simulation of the pulsed laser generation in Nd:YAG/Cr:YAG/Ba(NO₃)₂ microchip lasers for two different lengths of the laser cavity. Raman generation of two Stokes pulses was observed in both cases. Dependences of the pulse repetition rate, the pulse energy, the pulse duration, and the transverse spatial distribution of the beam intensity on the pump power were recorded. The real-time dynamics of the development of laser and Stokes pulses was registered and simulated numerically. Pulses of the 2nd Stokes radiation at the 1.37 μm wavelength with a duration as short as 100 ps and an energy of 5 μJ were obtained for the case of the short-cavity laser. For the long-cavity laser, the pulse durations were 3 times longer and the energy remained unaltered. It has also been found that the two lasers have very different dependences of the waist diameter of the Stokes beam on the pump power. On the basis of the developed model, a numerical simulation was carried out. There was a very good agreement between the calculated and experimental pulse energies and pulse dynamics. The influence of the four-wave mixing process on the 1st and 2nd Stokes energetics and temporal dynamics were analyzed.

In Sect. 5, some conclusions are presented.

2 Experimental setup

The optical scheme of the microchip laser is shown in Fig. 1. The laser crystal was a composite diffusion-bonded Nd:YAG/Cr:YAG crystal. The concentration of neodymium was 1.15 %. The concentration of chromium ions was chosen such that the initial transmission of the 1.4-mm Cr:YAG crystal was 70 %. The diffusion-bonded crystal permits decreasing the number of optical surfaces, and thus decreasing the passive optical loss, which is important for Raman conversion of laser radiation. The crystal was mounted on a copper heatsink with a temperature stabilized at 20 °C by a thermoelectrical cooler.

Fig. 1

Laser scheme

A fiber-coupled 30 W diode stack emitting at 805 nm was used as a pumping source. In our experiments, in order to decrease the thermal load on the crystal, the diode current was modulated at 200 Hz, the pulse duration being 0.5 ms, which corresponds to an off-duty ration of 1/10. The diode current amplitude was fixed at 37 A to hold constant the spectral properties of the pumping radiation. The pump power was varied by an attenuator consisting of a polarizer and a rotating quarter-wave plate. The pump radiation was transported by a 200 μm fiber with an NA of 0.22, and then it was collimated by an aspherical lens of focal length 11 mm and finally focused into the crystal by an aspherical lens of focal length 15 mm. The pumping beam diameter in the laser crystal was about 280 μm.

The input (rear) mirror was deposited directly on the surface of the laser crystal and provided a transmission coefficient of about 90 % for the pump, and a reflectivity for the laser radiation (1.064 μm) of over 98 %. The intracavity mirror was deposited directly on the Cr:YAG exit face of the crystal. It provided a reflectivity of over 98 % at the 1st and 2nd Stokes wavelengths (1.197 and 1.37 μm, respectively, in the barium nitrate crystal) and a reflectivity of 13 % at the laser wavelength.

A 2-mm thick barium nitrate crystal was used as a Raman active medium. It had antireflection coatings on both facets with a residual reflectivity of less than 1 % in the 1.0–1.4 μm wavelength range. These coatings also protected it from atmospheric moisture. The output mirror was plane having reflectivities of 98.7, 98.9, and 84.3 % at the laser and the 1st and 2nd Stokes wavelengths, respectively.

The output power of the laser and Stokes radiation was measured by a powermeter with thermoelectric sensor (Coherent PM3Q). Optical signals were detected by InGaAs and Silicium p-i-n photodiodes with a bandwidth of 12 GHz and analyzed by a Lecroy SDA 11000 digital real-time oscilloscope with a bandwidth of 11 GHz. Recording was carried out simultaneously for all spectral components with triggering by a 2nd Stokes pulse. The response of the oscilloscope system to a “delta”-pulse was 66 ps for the 2nd Stokes recording channel and 130 ps for the laser and 1st Stokes recording channels. The transverse spatial distributions of the beam intensities were recorded by a silicon CCD camera with a square pixel size of 4.4×4.4 μm (Fig. 2). In power and spatial measurements, the laser and Stokes beams were separated by a set of selective mirrors.

Fig. 2

Scheme of recording the transverse intensity distribution pattern at the output mirror

3 Model

Since in this work we investigate the dynamics of pulse generation of an SRS laser based on an Nd³⁺:YAG crystalline active medium, in formulating the laser rate equations of such lasers, it is necessary to take into account the features of stimulated emission in this crystal at a wavelength of 1.064 μm typical of neodymium ions in YAG. In Nd³⁺:YAG crystals, transitions with emission of radiation at the wavelength of 1.064 μm in two channels from two levels R₁ (11 423 cm⁻¹) and R₂ (11 507 cm⁻¹) of the upper laser multiplet ⁴F_3/2 down to two levels of Y₂ (2028 cm⁻¹) and Y₃ (2110 cm⁻¹) of the lower multiplet ⁴I_11/2, respectively, take place [26]. Both transitions have slightly different radiation wavelengths (1.0644 and 1.06415 μm), and the distance between them (≈2 cm⁻¹ [26, 27]) is about three times smaller than the stimulated emission bandwidth (≈6.5 cm⁻¹ [26, 27]) of neodymium ions in YAG, which makes them virtually indistinguishable in passively Q-switched lasers. Transitions in the channels R ₁→Y ₂ and R ₂→Y ₃ have different stimulated emission cross-sections, namely $\sigma_{R_{1} \to Y_{2}} =1.2\times 10^{-18}~\mathrm{cm}^{-3}$ and $\sigma_{R_{2} \to Y_{3}} =6.5\times 10^{-18}~\mathrm{cm}^{-3}$ [26, 27]. In the quasistationary regime of laser generation, when the beam intensity and the level populations in both laser multiplets change much less rapidly than the relaxation of activator atoms (here Nd) within the multiplets (the so-called thermalization [28]), the transitions in these channels can be considered as transitions in one channel (two-level scheme) with the “effective” cross-section averaged over the multiplets, σ _eff≈3.3×10⁻¹⁸ cm⁻³. For neodymium ions in crystalline active media, the thermalization times are of subnanosecond scale (≈0.7 ns in the Nd³⁺:YAG crystal [8]; ≈0.3 ns and ≈0.6 ns in the Nd³⁺:LSB crystal [4, 12] for the lower and upper multiplets of generation, respectively). Therefore, in passively Q-switched microchip and mini lasers based on Nd-doped active media, where the pulse duration can be in the range from ≈0.1 ns (or even less) to a few nanoseconds [6–19], because of the wide difference between the cross-sections $\sigma_{R_{1} \to Y_{2}}$ and $\sigma_{R_{2}\to Y_{3}}$ the interpretation of stimulated emission as occurring in one channel with an “effective” stimulated emission cross-section is no longer adequate to the real physical processes. In such lasers, the transitions in both channels of stimulated emission have different temporal characteristics of the replenishment and depletion dynamics of their level populations and, therefore, make different contributions to the increase in the laser (1.064 μm) intensity. Thus, in the model of passively Q-switched microchip and mini lasers based on Nd-doped YAG crystals, the equation for the laser (fundamental) wave amplitude and the equations for the active medium should take into account the transitions in both channels of stimulated emission independently.

We shall approximate the spatial distribution of the laser (1.064 μm) radiation intensity in the cavity by a TEM₀₀-like Gaussian beam distribution. As will be seen from the experimental data for the lasers studied in this work, this approximation is valid. The thickness of the layer of the composite crystal doped with neodymium ions that absorbs almost the whole of the pump radiation, is a few millimeters, and at such short lengths the radius of the transverse intensity distribution, r _L , varies insignificantly. This allows us to consider it approximately to be constant ($r_{L} \approx r_{L}^{(\mathrm{am})}$) in this layer of the active medium. Also, because the layer of the composite crystal doped with chromium serving as a saturable absorber is thin (in the lasers under investigation the length of this layer is 1.4 mm), the radius of the transverse laser intensity distribution in this layer is practically constant ($r_{L} \approx r_{L}^{(\mathrm{sa})}$). Within the framework of such approximations, on the basis of the quantum model of mini and microchip lasers with intracavity Raman conversion [30] and the semiclassical model proposed in [25] for the Nd³⁺:LSB/Cr⁴⁺:YAG-based two-Stokes laser with stimulated emission only in one channel (which is true for the active medium Nd³⁺:LSB), for the laser with stimulated emission in two independent channels, we obtain the following rate equation for the laser (fundamental) wave amplitude, A _L (t):

(1)

where c is the speed of light in vacuum, $n_{R_{2}}(r,t)$, $n_{R_{1}}(r,t)$, $n_{Y_{3}}(r,t)$, $n_{Y_{2}}(r,t)$, and n _a (r,t) are the 2D-densities of activator atoms in the active medium at the upper and lower laser levels in the upper (R ₂ and R ₁) and lower (Y ₃ and Y ₂) laser multiplets and the 2D-density of absorbing atoms in the ground state in the absorber, respectively,

$$ n_{i}(r,t) =\int_{L^{(\mathrm{am})}(L^{(\mathrm{sa})})}n_{i}(z,r,t)\,dz ,$$

(2)

the subscript i stands for R ₂, R ₁, Y ₃, Y ₂, or a, respectively, L ^(am) and L ^(sa) are the lengths of the layers of the Nd³⁺:YAG/Cr⁴⁺:YAG crystal doped with neodymium and chromium ions, respectively; L _L is the optical length of the cavity for the laser (1.064 μm) radiation, $L_{S_{1}}$ is the optical length of the cavity for the 1st Stokes radiation, σ _13,L and σ _24,L are the ground- and the excited-state cross-sections of fundamental wave absorption in the saturable absorber, respectively; n _a0=N _a L ^(sa), N _a is the 3D-density of absorbing atoms in the absorber layer; L ^(Rm) is the Raman medium length; N is the density of Raman-active atoms (molecules) in the Raman medium; (∂α/∂q) is the derivative of the Raman-medium polarizability with respect to the space coordinate; ω _L and $\omega_{S_{1}}$ are the cyclic frequencies of the laser and the 1st Stokes wave, respectively; μ _L and $\mu_{S_{1}}$ are the refractive indices of the Raman medium at the laser (fundamental) and the 1st Stokes wavelength, respectively; $Q_{L,S_{1}}(t)$ is the normalized amplitude of collective vibrations arising in the Raman medium in the process of SRS-conversion of the laser wave into a 1st Stokes wave; R _L,in and R _L,out are the reflection coefficients of the rear and the output mirror at the laser wavelength, respectively; α _L is the internal loss of the laser wave; β _sp is the fraction of spontaneously emitted photons contributing to the laser wave; τ _ul is the lifetime of activator atoms at the upper laser levels (here they are assumed to be equal). The product A ^∗(t)A(t) is the total number of laser or Stokes photons in the laser cavity. The first term on the right-hand side of (1) describes the stimulated emission in the active medium. The second and the third terms describe the ground- and excited-state absorption of the laser wave in the absorber. The fourth term on the right-hand side of (1) describes the intracavity SRS-conversion of the fundamental wave into a 1st Stokes wave. The fifth term corresponds to the loss, and the sixth term describes the contribution of spontaneous emission in the active medium to the fundamental wave.

The equation for the 1st Stokes wave amplitude is of the form [25]:

(3)

where $L_{S_{2}}$ is the optical length of the cavity for the 2nd Stokes radiation, $\omega_{S_{2}}$ is the cyclic frequency of the 2nd Stokes wave, $\mu_{S_{2}}$ is the refractive index of the Raman medium at the 2nd Stokes wavelength, $Q_{S_{1},S_{2}}(t)$ is the normalized amplitude of collective vibrations arising in the Raman medium in the process of SRS-conversion of the 1st Stokes wave into a 2nd Stokes wave, $\alpha_{S_{1}}$ is the internal loss of the 1st Stokes wave, $R_{S_{1},\mathrm{in}}$ and $R_{S_{1},\mathrm{out}}$ are the reflection coefficients of the intracavity and output mirrors at the 1st Stokes wavelength, respectively.

For the 2nd Stokes wave amplitude, the rate equation is of the form [25]:

(4)

where $\alpha_{S_{2}}$ is the internal loss of the 2nd Stokes wave, $R_{S_{2},\mathrm{in}}$ and $R_{S_{2},\mathrm{out}}$ are the reflection coefficients of the intracavity and output mirrors at the 2nd Stokes wavelength, respectively.

For the amplitudes of collective vibrations $Q_{L,S_{1}}(t)$ and $Q_{S_{1},S_{2}}(t)$ arising in the Raman medium in the process of SRS-conversion of the laser wave into a 1st Stokes wave and of the 1st Stokes wave into a 2nd Stokes wave, the rate equations are [25]:

(5)

and

(6)

where

$$ \delta_{L,S_{1};S_{1},S_{2}} = \frac{\sin ( \Delta k_{L,S_{1};S_{1},S_{2}}L^{(\mathrm{Rm})} )}{2\Delta k_{L,S_{1};S_{1},S_{2}}L^{(\mathrm{Rm})}},$$

(7)

$\Delta k_{L,S_{1};S_{1},S_{2}} = ( k_{L} - k_{S_{1}} ) - (k_{S_{1}} - k_{S_{2}} )$, $k_{L(S_{1},S_{2})}$ is the wave number of the laser (1st or 2nd Stokes) wave, $\varSigma_{L,S_{1};L,S_{1}}^{0}$, $\varSigma_{S_{1},S_{2};S_{1},S_{2}}^{0}$ and $\varSigma_{L,S_{1};S_{1},S_{2}}^{0}$ are the overlap integrals of the spatial distributions of the corresponding modes [25, 29], ω ₀ is the resonant cyclic frequency corresponding to the Raman shift (ħω ₀ is the Raman shift), m is the mass of Raman-active atoms, T ₂ is the Raman medium dephasing time (transverse relaxation time). For the cases where the transverse intensity distribution can be approximated by Gaussian functions, the overlap integrals $\varSigma_{L,S_{1};L,S_{1}}^{0}, \varSigma_{S_{1},S_{2};S_{1},S_{2}}^{0}$ and $\varSigma_{L,S_{1};S_{1},S_{2}}^{0}$ in (5) and (6) are [25, 29]

$$ \everymath{\displaystyle }\begin{array}{@{}l}\varSigma_{L,S_{1};L,S_{1}}^{0} =\frac{2}{\pi (r_{L}^{(\mathrm{Rm})} )^{2} + \pi ( r_{S_{1}}^{(\mathrm{Rm})} )^{2}},\\ \noalign {\vspace {6pt}}\varSigma_{S_{1},S_{2};S_{1},S_{2}}^{0} = \frac{2}{\pi (r_{S_{1}}^{(\mathrm{Rm})} )^{2} + \pi ( r_{S_{2}}^{(\mathrm{Rm})} )^{2}},\\ \noalign {\vspace {6pt}}\varSigma_{L,S_{1};S_{1},S_{2}}^{0} \\ \noalign {\vspace {6pt}}\quad {=}\,\hbox{\small $\dfrac{2}{\!\pi r_{L}^{(\mathrm{Rm})}r_{S_{2}}^{(\mathrm{Rm})} + \frac{1}{2}\pi ( r_{S_{1}}^{(\mathrm{Rm})})^{2}( r_{L}^{(\mathrm{Rm})} / r_{S_{2}}^{(\mathrm{Rm})}+ r_{S_{2}}^{(\mathrm{Rm})} /r_{L}^{(\mathrm{Rm})} )\!}$}.\end{array}$$

(8)

In (5) and (6), the terms describing the conversion of the laser wave into a 1st Stokes, the conversion of the 1st Stokes wave into a 2nd Stokes wave, and the multiwave mixing, contain the phenomenological factors $\xi_{LS_{1}}$, $\xi_{S_{1}S_{2}}$, and ξ _mw introduced [25, 29] to take into account the difference of the real transverse distributions of the beam intensities from the ideal Gaussian TEM₀₀-like distributions. We also introduced into (5) and (6) the phenomenological factor of Raman gain efficiency ξ _g (ξ _g ≤1) in order to take into account the possible decrease in the Raman gain due to the broad frequency bands of the fundamental and the Stokes waves typical of mini and microchip lasers [25, 29]. As can be seen from (5) and (6) for the amplitudes of collective vibrations of the Raman medium, the coupling factor, $\delta_{L,S_{1};S_{1},S_{2}}$, determines the intensity of multiwave mixing and its contribution to the initiation and amplification of the 2nd Stokes wave, and accordingly, to the attenuation of the 1st Stokes wave [25, 29]. The parameters entering into the right-hand side of (1)–(6) define the 1st Stokes and the 2nd Stokes gains, $g_{S_{1}}$ and $g_{S_{2}}$, respectively, in accordance with the known expression [30]

$$ g_{S_{1(2)}} =\frac{4\pi^{2}T_{2}\omega_{S_{1(2)}}}{\mu_{L(S_{1})}\mu_{S_{1(2)}}c^{2}m\omega_{0}}N \biggl( \frac{\partial \alpha}{\partial q} \biggr)^{2}.$$

(9)

For the 2D-concentrations of activator atoms at the levels of the upper and lower laser multiplets, the rate equations can be written using the approach discussed in [25], complementing it for the case of transitions in the two channels of stimulated emission. The equations are, respectively, of the following form:

(10)

(11)

(12)

(13)

(14)

where n _Y (r,t) is the total 2D-density of activator atoms at all levels of the lower multiplets except for the laser levels Y ₂ and Y ₃, $\eta_{R_{1}}$ is the fraction of nonradiative transitions of activator atoms from the upper pump bands to the laser level R ₁ in the upper multiplet, τ _u and τ _l are the thermalization times of activator atoms in the upper and lower laser multiplets, respectively, τ _ul is the luminescent lifetime of activator atoms in the upper laser levels, τ _lg is the lifetime of neodymium ions in the lower laser multiplet, $f_{R_{1(2)}}$ and $f_{Y_{2(3)}}$ are the Boltzmann factors of the upper R ₁₍₂₎ and lower Y ₂₍₃₎ laser levels, respectively, η _eff is the efficiency of pump absorption in the active medium, ħω _P is the pump photon energy, W _P is the pump power, and $r_{P}^{(\mathrm{am})}$ is the effective Gaussian radius of the pump beam in the active medium.

The saturation and recovery dynamics of the absorber based on Cr⁴⁺-doped crystals is described from the viewpoint of the 4-level model by the rate equation for the 2D-concentration of chromium ions in the ground state, n _a (r,t), [32–34]

(15)

where τ ₂₁ is the excited-state lifetime of chromium ions.

4 Results

Before studying the coupled-cavity lasers, we tried to obtain two-Stokes generation of lasers with a cavity common for the laser and the Stokes waves. In this case, both the input and the output mirror had a high reflectivity in the range of wavelengths from 1.06 to 1.4 μm. Falling within this range is a rather strong line (1.34 μm) of amplification of neodymium ions in the garnet. As a result, in the active medium, radiation was generated not only at the wavelength of 1.064 μm, but also at the wavelength of 1.34 μm. In so doing, for the 1.064 μm line Q-switching was observed and for the line of 1.34 μm self-oscillations took place. The energy of the 1st Stokes pulses was very low thereby, and 2nd Stokes pulses were not observed at all. This result confirmed the need to separate the laser and the Stokes resonators using the scheme of the coupled-cavity laser to make the process of SRS-conversion of laser pulses to 2nd Stokes pulses effective.

We have studied the pulse oscillation of two coupled-cavity microchip lasers with different lengths. In the first laser (we will refer to it as the short-cavity laser, #1), the air gaps between the Raman crystal and the mirrors forming the cavity for Stokes waves (see Fig. 1) are of 0.75 mm. In the second laser (we will refer to it as the long-cavity laser, #2), these gaps are of 2 mm. For these lasers, we measured the average output powers of the laser wave and of two Stokes waves, the pulse repetition rates, the pulse durations and shapes for the pump power in the range from 5 to 11 W. In this pump power range, transverse intensity distribution patterns for the laser and the 1st Stokes beam at the output mirror were also recorded.

Figures 3 and 4 show the dependences of the average output powers of the laser and two Stokes waves and the pulse repetition rates on the pump power for lasers #1 and #2, respectively. As can be seen, the average output power of the 2nd Stokes wave reaches 11 and 9 mW at pulse repetition rates of 24 and 20 kHz for lasers #1 and #2, respectively.

Fig. 3

Average output power (a) and pulse repetition rate (b) of the laser #1 (short-cavity laser)

Fig. 4

Average output power (a) and pulse repetition rate (b) of the laser #2 (long-cavity laser)

The transverse intensity distribution patterns for the laser and the 1st Stokes beam at the output mirror (for the scheme see Fig. 2) were recorded for all pump powers at which the average output powers, the pulse repetition rates, and the pulse shapes were measured. As examples, we present these profiles in Fig. 5 for laser #1 and Fig. 6 for laser #2. All recorded transverse intensity distribution patterns were approximated by Gaussian TEM₀₀-like distributions with the corresponding radii. Figure 7 shows the dependence of these radii for the laser and the 1st Stokes beams on the pump power. Figure 7 presents also the spread of values of the calculated radii obtained at certain variations of processing the experimental transverse intensity distribution patterns. As can be seen, for the Stokes beam the spread of values of the calculated radii are greater than those for the laser beam. We attribute this to the fact that in recording the transverse intensity distribution of the Stokes beam we used a film filter (see Fig. 2) for the spectral separation of the 1st Stokes wave, which led to the formation of interference distortions in the transmitted beam. In recording the transverse intensity distribution patterns for the laser beam, we used for the spectral separation of the 1.064 μm component a mirror that causes no interference distortions in the intensity distribution. Typical of both lasers is the fact that the waist radius of the laser beam at the output mirror decreases with increasing pumping. This can be considered to be natural, because due to the positive thermolens induced in the YAG-based active medium the cavity for the fundamental radiation is stable and, therefore, the focal length of the thermolens decreases with increasing absorbed pump power. At the same time, the Stokes radiation cavity contains a barium nitrate crystal in which, due to the Raman process a thermolens with a negative focal length is induced. As a result, the SRS-resonator is unstable, which affects the transverse intensity distribution of Stokes pulses. We think that this feature may be connected with the observed differences in diameters of the Stokes beams and their dependences on the pump power for lasers with a different length of the Raman cavity. As is seen from Fig. 7, in the case of the short-cavity laser (#1), the dependence of the 1st Stokes beam radius on the pump power is similar to that of the laser beam, but the value of the 1st Stokes beam radius exceeds by ≈15–30 μm the radius of the laser beam throughout the pump power range. In the case of the long-cavity laser (#2), the dependence of the 1st Stokes beam radius on the pump power is entirely different from that of the laser beam radius. For laser #2, the 1st Stokes beam radius increases almost linearly and at a pump power higher than ≈8 W it exceeds the radius of the laser beam. Thus, a change in the Raman cavity length due to the variation of the air gap lengths leads to marked changes in the radius of the 1st Stokes output beam. The slight increase and the subsequent decrease in the laser beam radius are associated with the increase in the optical power of the thermolens with increasing pump power. For the laser beam, the difference in the cavity length between the short- and the long-cavity lasers is about 15 %, while for the Stokes beams this difference is about 50 %. Therefore, the pump power dependence of the laser beam radius is little affected by a change in the laser length, while for the Stokes beam radius it changes radically.

Fig. 5

Laser (a) and 1st Stokes (b) beam profiles of the short-cavity laser (#1). The pump power is 10.3 W

Fig. 6

Laser (a) and 1st Stokes (b) beam profiles of the long-cavity laser (#2). The pump power is 11.2 W

Fig. 7

Laser and Stokes beam radii vs. the pump power (W _P ) for lasers #1 and #2

Figures 8 and 9 show the experimental (dots) dependence of the energy and duration of the laser and Stokes pulses on the pump power for lasers #1 and #2, respectively. The pulse energies were determined as the ratio of the measured average radiation power at the corresponding wavelength to the number of pulses emitted in one second determined as the pulse repetition rate divided by the off-duty ratio of the pump (in our experiments the off-duty ratio was 10). At low pump powers, the errors in determining the average output power and the pulse repetition rate are greater than at high pumping levels. Therefore, the dependences of the laser and the Stokes pulse energy undergo high fluctuations, and then, with increasing pump power, they become more or less constant. Comparison of the two lasers shows that in the laser with a longer Raman cavity the pulse energy fluctuations are higher than in the laser with a shorter Raman cavity, which may be due to the greater loss of the Stokes radiation in the unstable plane Fabry–Perot resonator. At the same time, the average energy values of the laser and the Stokes pulses for both lasers are practically the same.

Fig. 8

Energy and duration of the laser and Stokes pulses vs the pump power for laser #1

Fig. 9

Energy and duration of the laser and Stokes pulses vs the pump power for laser #2

On the basis of the model described in Sect. 3 the energies of the output laser and Stokes pulses and their durations have been calculated. The laser parameters used in the calculations are presented in Table 1. The results of modeling are shown in Figs. 8 and 9 (solid lines). In the calculations, the waist radii of generated beams were given by fitting the experimental data. We have checked several functions to fit the data, and found out that polynomial + hyperbolic ones provide accurate and rather simple fitting suitable for numerical calculations. The radii (in micrometers) of the laser and the 1st Stokes beams for laser #1 are given by the expressions

(16)

(17)

and for laser #2 they are given by the expressions

(18)

(19)

where W _P is given in Watt. It was assumed that the radii of the 2nd Stokes beams are equal to the radii of the 1st Stokes beams, which is acceptable for a short Fabry–Perot resonator with a weak thermolens induced in the intracavity element (barium nitrate crystal). The possible differences between the radii of Stokes beams of the 1st and 2nd order will be taken into account by varying the products of the parameters $\xi_{g}\xi_{LS_{1}}\xi_{\mathrm{mw}}$ and $\xi_{g}\xi_{S_{1}S_{2}}\xi_{\mathrm{mw}}$ in (5) and (6) by comparing the results of numerical modeling with the obtained experimental data. It turned out that a good agreement with the experiment is obtained when the values of the products of these parameters are ≈0.9–0.95. Functions (16)–(19) are represented in Fig. 7 by solid lines. As can be seen from Figs. 8 and 9, the numerical results throughout the pump power range are in good agreement with the experimental data. One can notice that the calculated energies of the laser pulses (at the fundamental wavelength of 1.064 μm) are lower than the experimentally measured values. We think that the approximation of the transverse laser beam distribution by the TEM₀₀-like Gaussian function does not fully take into account the contribution to the stimulated emission in the regions of the active medium away from the axis of generated beams. Stimulated emission in these regions is responsible for the fact that the experimental laser pulse energy is higher than that calculated in the approximation of the TEM₀₀-like Gaussian beam.

Table 1 Parameters used in calculation

For both lasers, the dynamics of generation and succession of pulses from the laser one to the 2nd Stokes (i.e., the pulse shapes) were recorded for all values of the pump power at which the output powers were measured and the transverse intensity distribution patterns were recorded. Figures 10 and 11 show the dynamics of generation and succession of pulses for three values of the pump power for lasers #1 and #2, respectively. It should be noted that the start time of 2nd Stokes pulse generation falls on the time when in the cavity the radiation at the wavelength of 1.064 μm is still present and the 1st Stokes pulse reaches its maximum. Thus, the process of 2nd Stokes pulse generation begins at least as four-photon degenerate mixing. The effect of multiwave mixing on the generation of 2nd Stokes pulses will be discussed in more detail below.

Fig. 10

Shapes of the laser and Stokes pulses for laser #1 at a pump power of 10.3 W (a), 8.3 W (b), and 6.9 W (c)

Fig. 11

Shapes of the laser and Stokes pulses for laser #2 at a pump power of 10.3 W (a), 8.3 W (b), and 6.9 W (c)

As it seen, for both lasers the calculated pulse shapes agree well with the experimental ones. For pulses at the laser (1.064 μm) wavelength, the calculated rate of rise of intensity at the leading edge is also somewhat lower than the experimental one. This can be explained by the fact that the generation of the laser pulse begins in the TEM₀₀-mode, then the transverse intensity distribution of laser pulses widens with the beginning of SRS conversion causing enhanced losses for the TEM₀₀ mode. Accordingly, at the laser wavelength generation of transverse modes of higher order and larger area begins. But, in the calculations the radius of the transverse intensity distribution is assumed to be time-invariable and is determined as averaged by fitting the experimental data. As is seen from Figs. 10 and 11, the area under the numerically calculated shapes of laser pulses is somewhat smaller than the area under the experimental pulse shapes. This, as we see it, can serve as indirect evidence of the continuation of stimulated emission in the regions of the active medium away from the axis of generated beams. Since the area under the pulse shape is proportional to the pulse energy, the approximation of the transverse intensity distribution of laser pulses by the Gaussian distribution with a time-invariable radius can lead to underestimated values of calculated laser pulses energies. Comparison of the experimental and theoretical results for lasers #1 and #2 allows us to conclude that although the energy of the Stokes and laser pulses of these samples differs insignificantly, there is a considerable difference between the parameters of the transverse intensity distributions of the laser and the Stokes beams at the output mirror. It may be assumed that an increase in the energy of 2nd Stokes pulses can be achieved by controlling the transverse distributions of beams and their overlap in the Raman medium, e.g., by optimizing the focal lengths of the laser mirrors.

Figure 12 shows the calculated laser pulse shape and the time dependence of the inversion difference, $( n_{R_{1}}(r,t) - n_{Y_{2}}(r,t) ) - (n_{R_{2}}(r,t) - n_{Y_{3}}(r,t) )$, in the lower (R ₁→Y ₂) and the upper (R ₂→Y ₃) channels of generation of neodymium ions at the center of the beam for laser #1 at a pump power of 8.3 W. As would be expected in developing the model, this difference is time-variable. Consequently, in the process of generation of nanosecond pulses, the thermalization processes in the upper and the lower multiplets of neodymium ions in YAG cannot be considered as being either too slow or vice versa, almost instantaneous, and consequently, the stimulated emissions in both channels have different temporal characteristics of the replenishment and depletion dynamics of the population of the level involved. This makes the contributions of these channels to the laser radiation intensity markedly different; the more so that the cross-sections of the corresponding transitions (for Nd:YAG) differ by a factor of more than five. Thus, the simulation results point to the necessity of independent taking into account both stimulated-emission channels in designing Nd:YAG-lasers generating pulses with durations, in comparison with which the thermalization times in the upper and the lower generation multiplets are neither very long nor very short.

Fig. 12

Laser pulse shape and dependence of the difference of inversion in the lower and the upper stimulated-emission channels $( n_{R_{1}}(r,t) - n_{Y_{2}}(r,t) )- ( n_{R_{2}}(r,t) - n_{Y_{3}}(r,t) )$ at a beam axis (r=0) for laser #1 at a pump power of 8.3 W

To analyze the effect of multiwave mixing of the laser and the Stokes waves in the Raman medium on the generation and parameters of pulses, we calculated the output pulse parameters by varying the length of the SRS crystal (barium nitrate) from 1.6 to 2.2 mm at a constant distance between the mirrors. A change in the length of the crystal leads in turn to a change in the multiwave mixing factor (see (7)) from ≈0.012 to −0.11 in accordance with the dispersion law of barium nitrate [34]. The calculations were performed for laser #1 at a pump power of 8.26 W on the assumption that all the other laser parameters are invariable. With increasing length of the Raman crystal, the gaps inside the SRS cavity decreased so that the length of the latter remained unaltered. Figure 13 shows the calculated dependences of the pulse energy and duration on the length of the Ba(NO₃)₂ crystal. Figure 13 shows also the dependence of the multiwave mixing factor $\delta_{L,S_{1};S_{1},S_{2}}$ on the length of the Raman crystal. As is seen from Fig. 13, in a wide range of lengths of the Raman crystal where the absolute value of $\delta_{L,S_{1};S_{1},S_{2}}$ is large, the energy transfer from the 1st to the 2nd Stokes pulse is the most effective, and the calculated energies of 2nd Stokes pulses take the largest values (from ≈4 to ≈5 μJ). At the same time for a Ba(NO₃)₂ crystal of length from 1.6 to 1.7 mm, for which $\delta_{L,S_{1};S_{1},S_{2}}\approx 0$ and, therefore, there is no multiwave mixing in the Raman crystal, the energy of 2nd Stokes pulses reduces by nearly one-half, and their duration increases by a factor of about 1.5. The energy and duration of 1st Stokes pulses increases thereby almost three- and four-fold, respectively. Thus, the calculations show that multiwave mixing is important for the generation of 2nd Stokes pulses, and, controlling the coupling factor value of multiwave mixing by varying the Raman crystal length, we can effectively control the energy and time parameters of output Stokes pulses in coupled cavity lasers. This conclusion is consistent with the analysis of the influence of multiwave mixing on the pulse generation of Stokes pulses performed in [25].

Fig. 13

Calculated dependences of the energy and duration of pulses and the coupling factor of multiwave mixing for laser #1 on the length of the barium nitrate crystal. The pump power is 8.26 W

5 Conclusion

Experimental and theoretical studies of the pulse generation of two diode-pumped passively Q-switched Nd:YAG/Cr:YAG/Ba(NO₃)₂ coupled cavity microchip lasers with intracavity Raman-conversion of laser pulses into Stokes pulses of the first two orders have been made. The lasers had different lengths of the Raman the cavity due to the different lengths of air gaps (0.75 and 2 mm). The minimum duration of the 2nd Stokes pulses in the short-cavity laser was 100 ps and in the long-cavity laser 200 ps. The pulse energy was about 5 μJ in both cases. The peak radiation power at 1.37 μm was about 50 kW. The pulse repetition rate of both lasers reached 20–24 kHz. For the first time, the dynamics of two-Stokes pulse generation has been studied experimentally with a high temporal resolution. In a wide range of pump powers, the shapes of the laser and both Stokes pulses and transverse intensity distributions at the output mirror have been recorded. The transverse intensity distributions of Stokes pulses for the two lasers have different dependences of their radii on the pump power. At a pump power exceeding the threshold power, the radius of the Stokes beam in the laser with a long Raman cavity increases linearly and then exceeds the radius of the laser (fundamental) beam, while for the laser with a short Raman cavity the radius of the Stokes beam decreases but always exceeds the radius of the laser (fundamental) beam by 20–30 μm. We propose a model of such lasers that takes into account their generation features determined by the application of the coupled cavity scheme and the miniature design. In the model, for lasers with an Nd:YAG-active medium the transitions in two channels of stimulated emission at a wavelength of 1.064 μm with different cross-sections are taken into account. It has been shown that the account of both stimulated-emission channels is required for the correct description of the dynamics of lasers with an Nd:YAG-active medium that generate pulses with a duration comparable to the thermalization time of the activator atoms in the upper and the lower laser multiplets. The numerically simulated laser and Stokes generation dynamics, the pulse shapes, and their durations and energies are in good agreement with all experimental data obtained.