Stimulated Raman Scattering in Alkali-Earth Tungstate and Molybdate Crystals at Both Stretching and Bending Raman Modes under Synchronous Picosecond Pumping with Multiple Pulse Shortening Down to 1 ps

Abstract

Comparative investigation of characteristics of spontaneous and stimulated Raman scattering (SRS) in different alkali-earth tungstate and molybdate crystals at both high and low frequency anionic group vibrations is presented. It has been found that, among these crystals, the SrMoO₄ and SrWO₄ crystals are the most perspective for SRS generation on both stretching and bending modes of internal anionic group vibrations with the strongest SRS pulse shortening under synchronous laser pumping because of not only highly intense stretching mode Raman line for efficient primary extra cavity long-shifted SRS conversion but also the widest bending mode Raman line for the strongest SRS pulse shortening down to the inverse width of the widest Raman line (~1 ps) at secondary intracavity short-shifted SRS conversion. The strongest 26-fold pump pulse shortening down to 1.4 ps at the Stokes component with the combined Raman shift in the synchronously pumped extra cavity SrMoO₄ and SrWO₄ Raman lasers has been demonstrated. It was found that synchronously pumped cascade SRS with combined Raman shift is more efficient in the SrWO₄ crystal because the bending mode Raman line is more intense relative to the stretching mode Raman line than that in SrMoO₄.

1. Introduction

Alkali-earth tungstate and molybdate crystals belonging to a scheelite family were proposed as the most efficient active materials for stimulated Raman scattering (SRS) [1,2,3]. To date, such crystals were widely used in a great number of crystalline Raman lasers where the most intense Raman mode with the highest wavenumber of ${\nu }_1$ ~ 879–925 cm⁻¹ corresponding to totally symmetric stretching internal vibration of the crystal anionic group was utilized [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. These crystals can also be doped with rare-earth ions of Nd³⁺, Er³⁺, Tm³⁺ and Ho³⁺ [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53] to combine lasing and SRS properties in one active crystal for compact self-Raman laser operation in contrast to the most popular CVD-diamond [54,55] and Ba(NO₃)₂ [56,57] crystals which have an advantage of the highest Raman gain but cannot be activated by laser-ion doping. One more interesting property of the scheelite-type tungstate and molybdate crystals is the presence of two intense vibrational lines in their spontaneous Raman spectra: the most intense one corresponds to the above mentioned high-wavenumber (${\nu }_1$) stretching mode (A_g) but the second line has a low wavenumber of about ${\nu }_2$ ~ 324–336 cm⁻¹ and is the widest due to an overlap of two symmetric bending modes (A_g + B_g) of internal vibrations of the crystal anionic group [58]. In contrast to the single-pass SRS lasing in such crystals taking place only on the most intense ${\nu }_1$ Raman mode [4,5,6,9,11,35,39], utilization of the optical cavity for the Raman-active crystal can also stimulate the SRS lasing with the ${\nu }_2$ Raman shift. For the first time, it was demonstrated in reference [59] using a BaWO₄ crystal (${\nu }_1$ = 925 cm⁻¹, ${\nu }_2$ = 332 cm⁻¹) having the most intense ${\nu }_1$ and ${\nu }_2$ Raman lines among the alkali-earth tungstate and molybdate crystals because the Raman lines in BaWO₄ are the narrowest due to the highest size and mass of the Ba²⁺ cation in the row Ca²⁺→Sr²⁺→Ba²⁺ [2,3,5]. The SRS generation was obtained in the extra cavity BaWO₄ Raman laser with not only the primary ${\nu }_1$ shift but also with the secondary ${\nu }_2$ shift under nanosecond laser pumping at wavelengths of 0.53 μm [59] and 1.34 μm [60]. Later, single ${\nu }_2$-shifted SRS generation [61,62] and competition of the ${\nu }_1$ and ${\nu }_2$ lines [63,64] in BaWO₄ Raman lasers with intracavity nanosecond pumping at the wavelength of 1.06 μm were reported. Note that the bending-mode ${\nu }_2$ Raman line is several times wider than the stretching-mode ${\nu }_1$ Raman line. This property of the ${\nu }_2$ line has an importance for the transient regime of SRS but the ${\nu }_2$-shifted SRS radiation was generated in a quasi-steady-state regime [59,60,61,62,63,64].

Realization of transient SRS can help to strongly shorten the SRS pulse in comparison with the pump pulse. In the case of a transient single-pass SRS, the possibility of the radiation temporal shape control appears due to a delay of generating SRS pulse because the transient SRS gain depends on the pumping energy density and not the pumping intensity [65] in contrast to the quasi-steady-state SRS. However, picosecond pulse shortening at single-pass (${\nu }_1$-shifted) SRS in crystals [6,10,11,35] was weak (about 2 times). A more interesting method of pulse shortening in transient SRS is using the intracavity SRS that was theoretically predicted in work [66]. Such pulse shortening effect can be very strong because the SRS pulse should be shortened down to the inverse width of the spontaneous Raman spectrum line under intracavity mode-locked laser pumping that has a similar nature with ultra-short pulse formation in mode-locked lasers. The method was experimentally confirmed in different schemes of intracavity crystalline Raman lasers generating the first Stokes component [67,68] but lasing efficiency was low because of difficulty to make stable and efficient both fundamental and SRS mode-locked lasing in one common laser cavity. Another possible method of SRS pulse shortening is using the extra cavity synchronously-pumped mode-locked Raman laser which can be highly efficient. The SRS generation in such crystalline Raman laser takes place under pumping by a train of ultra-short laser pulses with a repetition period synchronized with a round-trip time of the Raman laser external cavity that was realized for the first time in a LiIO₃ crystal in work [69], where the possibility of SRS pulse shortening by several times using the Raman laser cavity length detuning have been shown. Such pulse shortening effect is produced by a combination of group velocity walk-off and strong pump pulse depletion [70]. In recent years, this method was tested on many SRS-active crystals such as CVD-diamond [71,72,73,74,75,76], KGd(WO₄)₂ [77,78,79], LiNbO₃ [80] and BaWO₄ [81,82]. As noted above, the last crystal (BaWO₄) has two (${\nu }_1$ = 925 cm⁻¹ and ${\nu }_2$ = 332 cm⁻¹) intense vibrational Raman lines. Consequently, for the first time, in a synchronously pumped system with the BaWO₄ crystal in a linear external cavity [81], picosecond SRS generation was observed as a cascade process of the primary extra cavity Raman conversion on the most intense ${\nu }_1$ Raman line and the secondary intracavity Raman conversion on the widest ${\nu }_2$ Raman line. By controlling the external cavity length, at the (${\nu }_1+{\nu }_2$)-shifted 1227-nm Stokes wavelength, the 12-fold pulse shortening down to 3 ps that is equal to the inverse width of the widest ${\nu }_2$ Raman line has been achieved. This can be explained by the theory of ultra-short SRS pulse formation at intracavity pumping [66] taking into account that the highly shortened (${\nu }_1+{\nu }_2$)-shifted SRS pulses were intracavity pumped by the ${\nu }_1$-shifted SRS pulses. Using the optimized ring external cavity configuration [82] allowed to achieve the synchronously pumped BaWO₄ Raman laser operation with high efficiency (up to 39%) and high output pulse energy (up to 53 nJ) of the SRS radiation with the combined (${\nu }_1+{\nu }_2$) Raman shift and strong pulse shortening (down to 3 ps) increasing the output peak power up to 9.6 kW at 1227 nm which was even higher than the pump peak power (6.1 kW) at 1063 nm.

In the present work, in order to achieve stronger pulse shortening, we focus on other alkali-earth tungstate and molybdate crystals besides BaWO₄ having very similar vibrational structure with the intense ${\nu }_1$ and ${\nu }_2$ Raman lines [3]. The ${\nu }_2$ line characteristics of these crystals are unknown but the ${\nu }_2$ line width seems to be wider than for BaWO₄ having the narrowest Raman lines. Therefore, the pulse shortening effect in these crystals can be stronger. Thus, we present comparative investigation of the characteristics of spontaneous and stimulated Raman scattering in different alkali-earth tungstate and molybdate crystals at both high and low frequency anionic group vibrations. We demonstrate possibilities of the strongest 26-fold pulse shortening down to 1.4 ps at the (${\nu }_1+{\nu }_2$)-shifted Stokes component in the synchronously pumped extra cavity Raman lasers based on SrMoO₄ and SrWO₄ crystals.

2. Theoretical Approach

It is well-known that amplification of a Stokes SRS field under an undepleted pumping condition is described by the system of coupled equations [65]

$$\begin{array}{l}\frac{\partial q^{\ast }(z,\tau )}{\partial \tau }+\Gamma q^{\ast }(z,\tau )=i{k}_1{E}_S(z,\tau )E_p^{\ast }(\tau ),\hfill \\ \frac{\partial E_S(z,\tau )}{\partial z}=-i{k}_2{q}^{\ast }(z,\tau )E_p(\tau ),\hfill \end{array}$$

(1)

where $k_1$ and $k_2$ are coupling coefficients including the Raman susceptibility, $E_p$ and $E_S$ are the pump and Stokes optical field amplitudes, respectively, $q$ is the vibrational field amplitude of the Raman-active medium, $\Gamma =1/{\tau }_R$ is the dumping coefficient of the vibrational amplitude, ${\tau }_R=1/\left(\pi c{\Delta \nu }_{\mathrm{R}}\right)$ is the dephasing time of Raman scattering of the active vibrational mode with a linewidth (FWHM) of ${\Delta \nu }_{\mathrm{R}}$, $z$ is the longitudinal coordinate, $\tau =t-z/u$ is the local time, running with the group velocity of light $u$ (the difference in the group velocities of SRS and pump radiations is neglected).

A solution of equations system (1) for quasi-steady-state SRS, if the pump pulse duration ${\tau }_p$ is much longer than the dephasing time ${\tau }_R$, is also well-known [65]. It can be easily found if we neglect the derivative in the first equation of the system (1):

$${\left|E_S(z,\tau )\right|}^2={\left|E_S(0,\tau )\right|}^2\exp \left(2k_1{k}_2{\left|E_p(\tau )\right|}^{2}z/\Gamma \right)={\left|E_S(0,\tau )\right|}^2\exp \left(g{I}_p(\tau )z\right),$$

(2)

where g is the steady-state Raman gain coefficient (in cm/W), $I_p\left(\tau \right)$ is the pump radiation intensity (in W/cm²), $g{I}_p\left(\tau \right)=2k_1{k}_2{\left|E_p\left(\tau \right)\right|}^2/\Gamma$ is the steady-state Raman gain factor (in cm⁻¹). Therefore, the quasi-steady-state SRS amplification has exponential growth with the increment [65]

$$G_0=g{I}_{p}L,$$

(3)

where $I_p$ is the average value of the pump radiation intensity, $L$ is the SRS interaction length. So, the quasi-steady-state (${\tau }_p\gg {\tau }_R$) single-pass SRS amplification is independent on the pump pulse duration ${\tau }_p$.

For arbitrary duration of the pump pulse, system (1) has been integrated only partially [65,83]:

$$E_S(z,\tau )=E_S(0,\tau )+\sqrt{k_1{k}_{2}z}{E}_p(\tau ){\displaystyle \underset{-\infty }{\overset{\tau }{\int }}\frac{E_p^{\ast }({\tau }^{\prime })E_S(0,{\tau }^{\prime }){\mathrm{e}}^{-\Gamma (\tau -{\tau }^{\prime })}{I}_1\left(\sqrt{4k_1{k}_{2}z[u(\tau )-u({\tau }^{\prime })]}\right)}{\sqrt{u(\tau )-u({\tau }^{\prime })}}d{\tau }^{\prime }},$$

(4)

where $I_1\left(x\right)$ is the modified first-order Bessel function and $u\left(\tau \right)={\int }_{-\infty }^{\tau }{\left|E_p\left({\tau }^{\prime }\right)\right|}^{2}d{\tau }^{\prime }$ is the pump energy parameter. The integral in expression (4) is unsolvable in general.

In the case of highly transient SRS with ${\tau }_p<{\tau }_R$, when the factor $\exp \left[-\Gamma \left(\tau -{\tau }^{\prime }\right)\right]$ in expression (4) can be replaced with unity, the solution was found [84] to be

$${\left|E_S(z,\tau )\right|}^2={\left|E_S(0,\tau )\right|}^2{I}_0^2\left(\sqrt{4k_1{k}_{2}zu(\tau )}\right).$$

(5)

Therefore, highly transient SRS gain increment G (at ${\tau }_p<{\tau }_R$) in the case of a rectangular pump pulse can be determined (at $z=L$ and $\tau ={\tau }_p$) as

$$G=\ln [I_0^2(\sqrt{2G_0{\tau }_p/{\tau }_R})]\approx 2\sqrt{2G_0{\tau }_p/{\tau }_R}-\ln (2\pi \sqrt{2G_0{\tau }_p/{\tau }_R}),$$

(6)

where we used the approximate equality assuming that the modified zero-order Bessel function is close to $I_0\left(x\right)\approx \exp \left(x\right)/\sqrt{2\pi x}$ which is valid for $x>1/4$. The second term in the approximate expression is essentially lower than the first term, so highly transient SRS amplification has growth close to exponential as for the quasi-steady-state case but the increment $G$ Equation (6) of this growth is essentially lower than $G_0$ Equation (3) due to not only a square root but also ${\tau }_p/{\tau }_R<1$. So, the SRS radiation intensity amplification at both quasi-steady-state and transient cases can be considered as $I_S=I_{S0}·\exp \left(G\right)$ where $I_{S0}$ and $I_S$ are input (or seed) and output intensities of the SRS radiation ($G\equiv G_0$ at the quasi-steady-state case). Expression (6) takes also into account that the quasi-steady-state SRS gain increment $G_0$ can be used as the pumping parameter even for transient SRS.

In work [85], the approximate solution, which is valid not only at highly transient case (${\tau }_p<{\tau }_R$) but at full transient SRS condition at ${\tau }_p<20{\tau }_R$, was found. However, this work was focused on high values of the pumping parameter $G_0$ (>10) to estimate single-pass transient SRS generation threshold. Therefore, the first term in the solvable integral Equation (4) was neglected.

Now, we need to estimate threshold of extra cavity transient SRS laser oscillation. The external cavity for the Raman-active medium is used to decrease the threshold value $G_{0th}$ of the pumping parameter $G_0$ and it can be even lower than unity balancing with the cavity losses in spite of transient SRS. Therefore, we need to solve Equation (4) more accurately.

As in reference [85], we use the approximation based on reducing the integrand to the exponential form to make the integral in expression (4) solvable. We use now more accurate approximation based on the asymptotic form of the modified n-order Bessel function $I_n\left(x\right)=\left[\exp \left(x\right)/\sqrt{2\pi x}\right]\left[1-\left(4n^2-1\right)/8x+\cdots \right]$ taking into account two first terms, that is, for $I_1\left(x\right)$ in Equation (4) we use $I_1\left(x\right)\approx \left[\exp \left(x\right)/\sqrt{2\pi x}\right]\left[1-3/8x\right]$ which is valid for $x>1/2$. In the obtained integrand, the coefficient at the exponent changes slowly and its dependence on the integration variable can be neglected. Instead of the Equation (4) in the case of a rectangular pump pulse, we obtain the equation for the SRS gain increment:

$$G\approx \ln \left[{\left|1+\frac{\sqrt[4]{2G_0{\tau }_p/{\tau }_R}}{\sqrt{8\pi }{\tau }_p}\left(1-\frac{3}{8\sqrt{2G_0{\tau }_p/{\tau }_R}}\right){\displaystyle \underset{0}{\overset{{\tau }_p}{\int }}{\mathrm{e}}^{-\Gamma ({\tau }_p-{\tau }^{\prime })/{\tau }_R+\sqrt{2G_0({\tau }_p-{\tau }^{\prime })/{\tau }_R}}d{\tau }^{\prime }}\right|}^2\right],$$

(7)

which is solvable. The solution is

$$\begin{array}{cc}\hfill G& \approx \ln [|1+\frac{\sqrt[4]{2G_0{\tau }_p/{\tau }_R}}{\sqrt{8\pi }{\tau }_p/{\tau }_R}\left(1-\frac{3}{8\sqrt{2G_0{\tau }_p/{\tau }_R}}\right)\{1-\exp \left(\sqrt{2G_0\frac{{\tau }_p}{{\tau }_R}}-\frac{{\tau }_p}{{\tau }_R}\right)\hfill \\ & {+\sqrt{\frac{\pi G_0}{2}}\left[\mathrm{erf}\left(\sqrt{\frac{{\tau }_p}{{\tau }_R}}-\sqrt{\frac{G_0}{2}}\right)+\mathrm{erf}\left(\sqrt{\frac{G_0}{2}}\right)\right]\exp \left(\frac{G_0}{2}\right)\}|}^2],\hfill \end{array}$$

(8)

which can be used for the rectangular pump pulse with arbitrary duration ${\tau }_p$ at $G_0{\tau }_p/{\tau }_R>1/8$.

In the case of single-pass quasi-steady-state SRS, the criterial condition of the SRS generation threshold was determined in Reference [86] as $G_{0th}\approx 25$ corresponding to conversion of the pump radiation into the Stokes SRS component of approximately 1%. For the transient case (${\tau }_p<20{\tau }_R$), a similar condition $G_{th}\approx 25$ can be used [11,85] implying that the threshold value of the pumping parameter $G_{0th}=g{I}_{p,th}L$ should be higher according to expressions (6) and (8).

Using the external cavity for the Raman medium allows to decrease the threshold of SRS generation but the threshold condition become dependent on ${\tau }_p$ even for quasi-steady-state SRS [23] because, in the optical cavity with a round-trip time ${\tau }_c$, the SRS oscillation takes place as long as the ${\tau }_p$-long pump pulse acts. If the SRS conversion is weak and depletion of pumping can be neglected, development of the extracavity Raman lasing at single-pass pumping can be defined by the expression [87]

$${\tau }_c\frac{d{I}_S}{dt}=\left(p\cdot G-K\right)I_S$$

(9)

where $p·G$ is the round-trip SRS gain increment, $G$ is the single-pass SRS gain increment described by expression (8), $p=1$ for a ring cavity, $p=2$ for a linear cavity at ${\tau }_p>{\tau }_c$, $K=ln\left(1/T_S{R}_S\right)$ is the round-trip loss increment defined by the losses of the cavity, $T_S$ is round-trip transmission of the intracavity medium including all intracavity losses per round-trip of the cavity (it is single-pass transmission for a ring cavity or double-pass transmission for a linear cavity), $R_S$ is reflection of the output coupler, ${\tau }_c=L_c/c$ or ${\tau }_c=2L_c/c$ is the round-trip time for a ring or linear cavity, respectively, $L_c$ is an optical length of the cavity. Integrating expression (9) with respect to time from zero up to ${\tau }_p$ (the case of the rectangular pump pulse), we obtain

$$I_S=I_{S0}\exp [(p\cdot G-K){\tau }_p/{\tau }_c]$$

(10)

and so by analogy with the single-pass SRS ($G_{th}\approx 25$), the extracavity SRS oscillation threshold became

$$(p\cdot G_{th}-K){\tau }_p/{\tau }_c\approx 25.$$

(11)

Taking into account all the cavity characteristics, we obtain

$$G_{th}\approx 25\frac{{\tau }_c}{{\tau }_{p}p}+\frac{1}{p}\ln \frac{1}{T_S{R}_S}$$

(12)

For example, for quasi-steady-state SRS (${\tau }_p\gg {\tau }_R$), using expression (3) we can express the threshold pump intensity [23]

$$I_{p,th}=\frac{25}{gL}\left(\frac{{\tau }_c}{{\tau }_{p}p}+\frac{1}{25p}\ln \frac{1}{T_S{R}_S}\right)$$

(13)

where the factor in the brackets is the cavity parameter which is lower than unity, so it decreases the threshold pump intensity $I_{p,th}$ in comparison with the single-pass SRS generation threshold of $I_{p,th}\approx 25/\left(gL\right)$. According to expression (13), decreasing the pump pulse duration ${\tau }_p$ in comparison with the cavity round-trip time ${\tau }_c$ leads to increasing the intensity $I_{p,th}$. At single pulse pumping with ${\tau }_p<{\tau }_c$, the cavity cannot help to decrease the SRS threshold, so it has the single-pass SRS threshold value $I_{p,th}\approx 25/\left(gL\right)$. Usually, the ${\tau }_c$ value exceeds a nanosecond and it is difficult to realize transient SRS in a cavity under pumping by a single pulse shorter than a nanosecond.

At the limit of ${\tau }_p\gg {\tau }_c$, from the expression (12), we obtain the lowest oscillation threshold condition

$$G_{th}\approx \frac{1}{p}\ln \frac{1}{T_S{R}_S}$$

(14)

usually used not only for Raman lasers but for all lasers because it presents the condition of balance between gain and losses in a laser.

Interesting possibility is opened using pumping by repetitive pulses with a repetition period synchronized with the cavity round-trip time ${\tau }_c$. The individual pump pulses can be shorter than ${\tau }_c$ and even ultra-short realizing transient SRS in a cavity. Such systems are synchronously pumped Raman lasers [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82]. If we use a train of ultra-short pulses with the pulse train duration of $t_p$ at the individual pulse duration of ${\tau }_p$, the extracavity synchronously pumped SRS oscillation threshold will be defined by expression (12) where ${\tau }_p$ should be replaced with $t_p$ and $p=1$ for both ring and linear cavities (at ${\tau }_p<{\tau }_c$ we have gain only in the forward direction) but the SRS gain $G$ (and its threshold value $G_{th}$ in expression (12)) is defined by ${\tau }_p$ as before by expression (8). Using mode locking for the pump laser, it is easy to obtain a very high value of $t_p$ (quasi-continuous-wave mode-locking) and even continuous-wave mode-locking ($t_p\to \infty$). Then, we fulfil the condition $t_p\gg {\tau }_c$ for the lowest SRS oscillation threshold and can apply expression (14) at $p=1$ for estimation of oscillation threshold in extracavity synchronously pumped quasi-continuous-wave mode-locked Raman lasers.

3. Raman Crystal Characterization

Alkali-earth orthotungstate and orthomolybdate crystals (MXO₄ where M = Ca, Sr, Ba; X = W, Mo) crystallize in the tetragonal scheelite-type structure of $C_{4h}^6$ space symmetry. The approximation of almost free anions (XO₄)²⁻ can be applied to these compounds because covalent bonds between Mo or W and four oxygen atoms are stronger than a bond between the full (WO₄)²⁻ or (MoO₄)²⁻ tetragonal anionic groups with the cation sublattice. This makes it possible to distinguish the so-called internal and external modes in the vibrational spectrum. The internal vibrations correspond to the oscillations inside the (WO₄)²⁻ or (MoO₄)²⁻ anionic groups with an immovable mass centre; the external or lattice phonons correspond to the motion of the cation and the rigid molecular unit [3]. This approximation allows to consider the most intense lines in spontaneous Raman spectrum as the anionic group internal vibrations splitting by a crystalline field. These most intense lines correspond to symmetric stretching and bending internal vibrations of anionic groups with high and low wavenumber of ${\nu }_1$ and ${\nu }_2$, respectively. The high wavenumber vibrational Raman mode at ${\nu }_1$ is the strongest corresponding to internal totally symmetric stretching vibration (A_g(${\nu }_1$) mode) of W-O or Mo-O bond in the anionic group [58]. The secondary Raman line at ${\nu }_2$ is less intense and broadened, corresponding to two overlapping modes A_g(${\nu }_2$) + B_g(${\nu }_2$) of symmetric bending of O-W-O or O-Mo-O bond [58]. The high wavenumber (${\nu }_1$) Raman mode characteristics of alkali-earth tungstate and molybdate crystals were studied in Reference [1,2,3]. It was found [3] that, for these crystals, excluding scheelite CaWO₄, values of the ${\nu }_1$ Raman line peak intensity (i.e., peak Raman scattering cross-section) are almost as large as in a Ba(NO₃)₂ crystal but values of integral Raman scattering cross-section at ${\nu }_1$ (which is determined by the ${\nu }_1$ Raman line peak intensity multiplied by its width $\Delta {\nu }_1$) are 2.5–3 times higher than for Ba(NO₃)₂ and only 1.6–2 times lower than for diamond (barium nitrate and diamond are the most popular Raman-active crystals having the highest Raman gain). Therefore, alkali-earth tungstate and molybdate crystals are the most perspective for both quasi-steady-state and transient regimes of SRS as well as diamond but they also have the secondary (${\nu }_2$) Raman line in comparison with diamond.

In order to study the ${\nu }_2$ Raman line characteristics and compare them with the ${\nu }_1$, we measured polarized spontaneous Raman scattering spectra of all alkali-earth tungstate and molybdate crystals with high spectral resolution. We used double monochromator “SPEX-Ramalog 1403” with the excitation by a 510.6-nm copper vapor laser with the spectral resolution of 0.5 cm⁻¹ and scanning step of 0.2 cm⁻¹.

Both the ${\nu }_1$ and ${\nu }_2$ lines in spontaneous Raman spectrum of alkali-earth tungstate and molybdate crystals are the most intense when the exciting light is linearly polarized along the crystal optical axis, that is, in the E‖c orientation. The comparative data measured for this orientation are presented in

Table 1. Spontaneous Raman scattering parameters of alkali-earth tungstate and molybdate crystals in E‖c orientation.

Crystal	${\mathit{\nu }}_{\mathbf{1}}$, cm−1	$\mathbf{\Delta }{\mathit{\nu }}_{\mathbf{1}}$, cm−1	${\mathit{I}}_{\mathbf{1}}$, *	${\mathit{\nu }}_{\mathbf{2}}$, cm−1	$\mathbf{\Delta }{\mathit{\nu }}_{\mathbf{2}}$, cm−1	${\mathit{I}}_{\mathbf{2}}$, *	${\mathit{I}}_{\mathbf{1}}\mathbf{/}{\mathit{I}}_{\mathbf{2}}$
CaWO4	910	6.9	18	334	9	10	1.8
SrWO4	921	2.7	41	336	8.7	12	3.5
BaWO4	925	1.6	64	332	3.8	27	2.4
CaMoO4	879	5.5	34	322	8	17	2
SrMoO4	888	2.6	51	327	10.5	11	4.8
BaMoO4	892	2.1	62	324	4	31	2

* Values of the Raman line peak intensities $I_{1,2}$ are presented in % with respect to the line peak intensity of the Raman-active mode (1333 cm⁻¹) of diamond with an equal length.

.

Values of the width ${\Delta \nu }_2$ of the ${\nu }_2$ Raman line, its peak intensity $I_2$ and the ratio $I_1/I_2$ of peak intensities of the ${\nu }_1$ and ${\nu }_2$ lines are the most important data for us here because we realize cascade SRS firstly on the most intense ${\nu }_1$ line and secondly on the widest ${\nu }_2$ line for the benefit of the strongest pulse shortening down to the inverse value of the widest linewidth $\Delta {\nu }_2$.

The ${\nu }_2$ line of the bending vibration for the (WO₄)²⁻ and (MoO₄)²⁻ anionic groups is slightly shifted along the tungstate (${\nu }_2$ = 332–336 cm⁻¹) and molybdate (${\nu }_2$ = 322–327 cm⁻¹) series, which is the same way as for the ${\nu }_1$ line of the stretching vibration for the tungstate (${\nu }_1$ = 910–925 cm⁻¹) and molybdate (${\nu }_1$ = 879–892 cm⁻¹) series. The ${\nu }_2$ wavenumber in the molybdates is lower because the unit cell is larger. Due to the overlap of two bending modes [A_g(${\nu }_2$) + B_g(${\nu }_2$)], the linewidth $\Delta {\nu }_2$ is essentially (2–4 times) wider than the linewidth ${\Delta \nu }_1$ except for CaWO₄ and CaMoO₄ having the highly broadened ${\nu }_1$ line.

In all the crystals, there is a correlation between the values $I_{1,2}$ and ${\Delta \nu }_{1,2}$ because an increase in the line intensities $I_{1,2}$ takes place due to an increase in the phase relaxation (dephasing) time ${\tau }_{1,2}=1/\pi c{\Delta \nu }_{1,2}$ at approximately equal values of the integral Raman scattering cross-section ($I_{1,2}·\Delta {\nu }_{1,2}$). The intensity $I_1$ of the high wavenumber line and its phase relaxation time ${\tau }_1=1/\pi c\Delta {\nu }_1$ have smooth increasing with cation replacement in the row Ca²⁺→Sr²⁺→Ba²⁺ for each anionic group. On the other hand, the low wavenumber line intensity $I_2$ and its phase relaxation time ${\tau }_2=1/\pi c\Delta {\nu }_2$ change only slightly with cation replacement Ca²⁺→Sr²⁺ but $I_2$ and ${\tau }_2$ increase significantly (more than 2 times) for Ba²⁺. This anomalous property of the ${\nu }_2$ Raman line can be caused by the same factors as its anomalously low difference in wavenumbers ${\nu }_2$ with cation replacement Ca²⁺→Sr²⁺→Ba²⁺ for each anionic group. It can be explained by compensation (for the ${\nu }_2$ Raman line) between the influence of growing mass and size of the cation with the influence of the Davydov splitting effect [3]. If the first is predominant for lower (<${\nu }_2$) wavenumber external vibrations decreasing the vibrational level energy with cation replacement Ca²⁺→Sr²⁺→Ba²⁺ (in this row the cation mass and size increase), the second is predominant for the highest (${\nu }_1$) wavenumber internal vibrations increasing the vibrational level energy in the row Ca²⁺→Sr²⁺→Ba²⁺ [3]. As a result, the SrWO₄ and SrMoO₄ crystals have the widest ${\nu }_2$ Raman lines as in CaWO₄ and CaMoO₄ and also have the highly intense ${\nu }_1$ Raman lines as in BaWO₄ and BaMoO₄. Hence, SrWO₄ and SrMoO₄ crystals are the most perspective having not only high Raman gain coefficients of g = 4.7 and 5.6 cm/GW at 1.06-µm pumping [11] (g is proportional to $I_1$) but also the widest ${\nu }_2$ Raman lines ($\Delta {\nu }_2$ = 8.7 cm⁻¹ and 10.5 cm⁻¹, respectively) allowing to obtain SRS pulse shortening down to $1/\pi c\Delta {\nu }_2\approx 1\mathrm{ps}$.

For comparison of the perspective crystals, we also need to take into account the ratio $I_1/I_2$ of peak intensities of the ${\nu }_1$ and ${\nu }_2$ lines. Figure 1 shows the high resolution polarized spontaneous Raman scattering spectra of the vicinities of the ${\nu }_1$ and ${\nu }_2$ lines in BaWO₄, SrWO₄ and SrMoO₄ crystals at exciting light polarization parallel (E‖c) and perpendicular (E$\perp$c) to the crystal optical axis.

It can be seen that in the BaWO₄ crystal, having the narrowest ${\nu }_1$ and ${\nu }_2$ Raman lines, the ${\nu }_2$ line is the most intense relative to the ${\nu }_1$ line (Figure 1a), that is, the ratio $I_1/I_2=2.4$ is the lowest. In the SrWO₄ crystal, the ${\nu }_2$ line is wider that resulted in its lower intensity relative to the ${\nu }_1$ line (Figure 1b) and so the ratio is intermediate $I_1/I_2=3.5$. And the lowest intense ${\nu }_2$ line relative to the ${\nu }_1$ line is in the SrMoO₄ crystal (Figure 1c) because the ${\nu }_2$ line is the widest at approximately equal widths of the ${\nu }_1$ line in SrMoO₄ and SrWO₄ and so the ratio is the highest $I_1/I_2=4.8$.

Thus, we can compare characteristics of the most perspective SrMoO₄ and SrWO₄ crystals having the widest ${\nu }_2$ line ($\Delta {\nu }_2$ ~ 10 cm⁻¹) for SRS pulse compression down to $1/\pi c\Delta {\nu }_2\approx 1\mathrm{ps}$ at cascade (${\nu }_1+{\nu }_2$)-shifted SRS conversion. The SrMoO₄ crystal has a higher intense ${\nu }_1$ line giving a higher Raman gain of 5.6 cm/GW at 1.06-µm pumping which is advantageous for the primary SRS conversion on the ${\nu }_1$ Raman line. However, the relative intensity of the ${\nu }_2$ line in this crystal is the lowest (the ${\nu }_2$ line intensity is 4.8 times lower than the ${\nu }_1$ line intensity) which is disadvantageous for the secondary SRS conversion on the ${\nu }_2$ Raman line. Contrariwise, the SrWO₄ crystal has a lower intense ${\nu }_1$ Raman line giving a lower Raman gain of 4.7 cm/GW at 1.06-µm pumping increasing the SRS oscillation threshold on the ${\nu }_1$ Raman line but the relative intensity of the ${\nu }_2$ line in this crystal is high (the ${\nu }_2$ line intensity is only 3.5 times lower than the ${\nu }_1$ line intensity) which is advantageous for the secondary SRS conversion on the ${\nu }_2$ Raman line. So, synchronously pumped cascade Raman laser oscillation in SrMoO₄ and SrWO₄ requires comparison at equal conditions. Such oscillation has been recently tested [81,82] for the BaWO₄ crystal having the highest Raman gain of 8.5 cm/GW at 1.06-µm pumping [5] and the lowest $I_1/I_2=2.4$ but the narrowest $\Delta {\nu }_2$ = 3.8 cm⁻¹. The possibilities of efficient cascade (${\nu }_1+{\nu }_2$)-shifted SRS conversion (up to 39%) and strong SRS pulse shortening down to the inverse width of the ${\nu }_2$ Raman line ($1/\pi c\Delta {\nu }_2=3\mathrm{ps}$) in BaWO₄ were demonstrated [82]. The cascade (${\nu }_1+{\nu }_2$)-shifted SRS generation in SrMoO₄ was also obtained [88].

4. Experimental Setup of the Synchronously Pumped Crystalline Raman Laser

The experimental setup of the synchronously pumped Raman laser system is shown in Figure 2. The laser system is the same as used for the BaWO₄ and SrMoO₄ crystal in the previous experiments [82,88]. The a-cut SrMoO₄ and SrWO₄ crystals were used as the Raman-active medium. Both crystals were grown by Czochralski technique at Prokhorov General Physics Institute of the Russian Academy of Sciences. The SrMoO₄ crystal had a shape of a Brewster-cut slab with the length of 17 mm and cross-section of 7 × 5 mm². The SrWO₄ crystal was also Brewster-cut with the length of 14 mm and cross-section of 10 × 7 mm². The crystal optical axis (c) was oriented horizontally for pumping by E‖c enabling to access the maximum intensities of the ${\nu }_1$ and ${\nu }_2$ Raman lines (Figure 1) and the minimum losses at the Brewster incidence.

As a source for synchronously pumped Raman laser, a laboratory-designed master oscillator power amplifier (MOPA) Nd:GdVO₄ laser system was used. The oscillator and amplifier were based on similar Nd:GdVO₄ slab crystal with the dimensions of 16 × 4 × 2 mm³ in the bounce geometry. The master oscillator lasing at the wavelength of ${\lambda }_p$ = 1063 nm generated in the continuously mode-locked regime using a semiconductor saturable absorber mirror (SESAM) with the modulation depth of 2% and non-saturable losses <2% [89]. The output continuous-wave train, consisting of 36 ps pulses at the repetition rate of 150 MHz, was amplified in the single-pass amplifier. This amplifier was pumped by a quasi-continuous (QCW) laser diode with the pulse duration of 500 µs at the repetition rate of 50 Hz. The QCW amplification of continuous wave mode-locked train was used to obtain long trains (t_p = 500µs) consisting of more than 70,000 picosecond pulses with smoothly varying intensity because the amplitude and temporal stability is crucial for the stable SRS generation. The energy of a single amplified pulse was up to 221 nJ.

The linearly-polarized pumping radiation with the beam spatial profile close to the fundamental transversal mode was focused into a Raman crystal centre with the spot radius of 35 × 30 µm² by an anti-reflection coated spherical lens SL with the focal length of 100 mm.

The external bow tie cavity for investigated Raman laser system consisted of two concave and two flat mirrors. The concave mirrors PM, which was used at a pumping mirror and M1 had the same curvature radius of 100 mm. Two OCs (OC1 and OC2) with different reflectivities were tested for generation with single (${\nu }_1$) and combined (${\nu }_1+{\nu }_2$) Raman shifts. The mirror reflectivities are summarized in

Table 2. Raman laser cavity mirrors reflectivities at pumping and Stokes wavelengths.

Mirror	R [%] 1063 nm	R [%] 1174/1178 nm	R [%] 1220/1227 nm	R [%] 1271/1279 nm	R [%] 1310/1321 nm
PM	1.3	98.9/98.9	99.4/99.4	98.1/97.1	73.6/34.0
M1	99.6	99.6/99.6	99.6/99.6	99.6/99.6	98.8/98.6
M2	99.8	99.9/99.9	99.8/99.8	99.5/99.3	89.4/47.0
OC1	99.6	87.4/88.7	85.1/80.1	51.5/68.9	66.7/63.3
OC2	99.8	99.9/99.4	96.1/96.3	78.6/66.1	23.3/16.0

. Note, that the Raman cavity was designed to compensate astigmatism.

For separation of individual generated Stokes components and pumping radiation, the long pass filters Thorlabs FEL1150, 1200, 1250 and 1300 (cut-on wavelength of 1150, 1200, 1250 and 1300 nm, respectively) were used. The generated SRS spectra were recorded by a spectrometer OceanOptics NIR512 operating in a wavelength range 850–1700 nm with a FWHM resolution of ~3 nm. The SRS output pulse energy was measured by a power detector Standa 11PMK-15S-H5. For determination of SRS pulse duration, the pulses were measured by a laboratory designed non-collinear second harmonic generation (SHG) autocorrelator based on a LiIO₃ crystal. Note that we assumed a Gaussian shape of the measured autocorrelation curves.

5. Experimental Results of SRS Lasing

The synchronously pumped SRS generation was successfully achieved in all tested crystals. The case of perfect synchronization corresponded to the lowest SRS generation threshold and the highest SRS radiation output energy. The SRS radiation in the ring cavity Raman laser was generated only in the forward direction because SRS oscillation threshold for the backscattering was essentially higher due to pumping by shorter pulse than the Raman-active crystal transit time [90].

Figure 3 demonstrates a comparison of the output SRS spectra for the 8-cm long BaWO₄ crystal studied earlier [82], for the 1.7-cm long SrMoO₄ [88] and 1.4-cm long SrWO₄ crystals studied here in the same configuration of the Raman laser without selective filtering of strong SRS components.

In the setup with the output coupler OC1, only the first Stokes components at the wavelength of 1179, 1174 and 1178 nm corresponding to Raman shift of ${\nu }_1$ = 925,888 and 921 cm⁻¹, were generated in BaWO₄, SrMoO₄ and SrWO₄, respectively (Figure 3, blue line). In the setup with the output coupler OC2, due to lower optical losses of the output coupler OC2 at 1174–1179 nm and 1220–1279 nm, the intracavity intensity of the ${\nu }_1$-shifted first Stokes SRS components were higher, so the additional Stokes components at the wavelength of 1227, 1220 and 1227 nm in BaWO₄, SrMoO₄ and SrWO₄, respectively, with a short wavenumber shift of ${\nu }_2$ (332, 327 and 336 cm⁻¹) from the ${\nu }_1$-shifted first Stokes component were also generated (Figure 3, red line).

Due to high intracavity intensity at the ${\nu }_1$ long-shifted first Stokes component, the $2{\nu }_1$ long-shifted second Stokes component at 1310 and 1321 nm in SrMoO₄ and SrWO₄, respectively, was also generated (Figure 3b,c) but it was weak because the output coupler OC2 had low reflectivity of 23–16% at 1310–1321 nm and even lower (13%) at 1323 nm preventing the 1323-nm $2{\nu }_1$-shifted SRS oscillation in BaWO₄ [82]. The 1323-nm $2{\nu }_1$-shifted SRS component in the synchronously pumped Raman laser based on the same (8-cm long) BaWO₄ crystal was efficiently generated in work [81] using higher reflectivity (>50%) output couplers at 1323 nm.

We can also see in Figure 3c one more additional Stokes component at 1279 nm oscillating only in SrWO₄. It was identified as the $2{\nu }_2$-shifted Stokes component from the ${\nu }_1$-shifted Stokes component. In comparison with SrMoO₄, the generation of this (${\nu }_1+{\nu }_2+{\nu }_2$) higher order Stokes component in SrWO₄ can be explained by that the vibrational ${\nu }_2$ Raman line has higher intensity relative to the vibrational ${\nu }_1$ Raman line in SrWO₄ ($I_1/I_2=3.5$, see Table 1) than in SrMoO₄ ($I_1/I_2=4.8$, see Table 1.

Figure 4 demonstrates the Raman laser radiation spectrum for the same 1.4 cm long SrWO₄ crystal at the output coupler OC2 using selective filtering of the strongest (1063, 1178 and 1227 nm) radiation components for registration of all weak SRS radiation components. We can see many additional SRS components with single ($+{\nu }_2$) and double ($+2{\nu }_2$) short Stokes shifts and even single ($-{\nu }_2$) short anti-Stokes shift from the single ($+{\nu }_1$) and double ($+2{\nu }_1$) long-shifted Stokes components.

All SRS radiation wavelengths possible in BaWO₄, SrWO₄ and SrMoO₄ crystals at 1063-nm pumping are calculated in

Table 3. SRS radiation wavelengths in BaWO₄, SrWO₄ and SrMoO₄ crystals at 1063-nm pumping.

SRS Radiation Wavelength	BaWO4	SrMoO4	SrWO4
λ1–2 = [λp−1 − (ν1 − ν2)]−1	1135 nm	1130 nm	1133 nm **
λ1 = [λp−1 − (ν1)]−1	1179 nm *	1174 nm *	1178 nm *
λ1+2 = [λp−1 − (ν1 + ν2)]−1	1227 nm *	1220 nm *	1227 nm *
λ1+1–2 = [λp−1 − (ν1 + ν1 − ν2)]−1	1268 nm	1256 nm	1265 nm **
λ1+2+2 = [λp−1 − (ν1 + ν2+ ν2)]−1	1279 nm	1271 nm	1279 nm *
λ1+1 = [λp−1 − (ν1 + ν1)]−1	1323 nm *	1310 nm *	1321 nm *
λ1+1+2 = [λp−1 − (ν1 + ν1 + ν2)]−1	1383 nm	1369 nm	1383 nm **

* experimentally realized with considerable conversion, ** experimentally realized with weak conversion.

where the experimentally realized SRS oscillation wavelengths are marked by asterisks.

If many additional short-shifted SRS components were generated in SrWO₄ because of the relatively intense vibrational ${\nu }_2$ Raman line ($I_1/I_2=3.5$, Table 1), it is strange that it was not generated in BaWO₄ having more intense ${\nu }_2$ Raman line in comparison with the ${\nu }_1$ Raman line ($I_1/I_2=2.4$, Table 1). It can be caused by difference of the crystal lengths. The SrWO₄ crystal was the shortest (1.4 cm) and the BaWO₄ crystal was the longest (8 cm). To verify that, we prepared a long SrWO₄ sample and tested it in an additional experiment in the same Raman laser setup. This Brewster-cut SrWO₄ crystal had a length of 3.6 cm. The output SRS radiation spectra are presented in Figure 5.

It can be seen that the only one short-shifted SRS radiation component was generated in the long SrWO₄ crystal (Figure 5), similarly as in the long BaWO₄ crystal (Figure 3a). It confirms that increasing the crystal length leads to the prevention of higher order short-shifted SRS component generation. It can be explained by a dispersion effect which is stronger in long media. And one more reason is a parametric coupling effect decreasing the oscillation threshold for higher order Stokes components in short crystals because the interaction length becomes lower than a coherence length of four-wave mixing of SRS components [30]. It is confirmed by generation of not only Stokes but also anti-Stokes (${\nu }_1-{\nu }_2$ and $2{\nu }_1-{\nu }_2$) short-shifted SRS components in SrWO₄ (Figure 4). Short-shifted anti-Stokes generation can be explained only by four-wave mixing process of parametric coupling of two photons of the long-shifted Stokes component with one photon from the short-shifted Stokes component and one photon from the short-shifted anti-Stokes component. Similarly, short-shifted second Stokes (${\nu }_1+2{\nu }_2$) generation can be explained not only by cascade SRS but also by four-wave mixing process of parametric coupling of one photon from the (${\nu }_1$) long-shifted Stokes component with two photons from the first (${\nu }_1+{\nu }_2$) short-shifted Stokes component and one photon from the second (${\nu }_1+2{\nu }_2$) Stokes component. For example, for SrMoO₄, using known Sellmeier equation [91], we can calculate wave mismatch of four-wave mixing generation of the (${\nu }_1+2{\nu }_2$)-shifted Stokes component at 1256 nm (Table 3) by the formula [30]

$$\Delta k=(n_{S0}+n_{S2}-2n_{S1})\cdot 2\pi {\lambda }_{S1}^{-1}+(n_{S0}-n_{S2})\cdot 2\pi {\nu }_2,$$

(15)

where $n_{S0}$, $n_{S1}$ and $n_{S2}$ are refractive indices at the Stokes components with the ${\nu }_1$, (${\nu }_1+{\nu }_2$) and (${\nu }_1+2{\nu }_2$) shifts, respectively, ${\lambda }_{S1}$ is the wavelength of the (${\nu }_1+{\nu }_2$)-shifted Stokes component. We obtain $\Delta k\approx 4.8{\mathrm{cm}}^{-1}$ corresponding to the coherence length [51] of $L_{coh}=\pi /\Delta k\approx 0.7\mathrm{cm}$ which is substantially shorter than the SrMoO₄ crystal length of 1.7 cm, so we had no (${\nu }_1+2{\nu }_2$)-shifted SRS generation at 1256 nm in the SrMoO₄ crystal (Figure 3b). On the contrary, the SrWO₄ crystal is shorter with a length of 1.4 cm being closer to the coherence length, so the (${\nu }_1+2{\nu }_2$)-shifted SRS generation at 1279 nm took place (Figure 3c).

On the other hand, dispersion effect in long media leads to significant time delay between different SRS components preventing generation of higher order SRS components too. It is not so strong in short crystals (~1–4cm) in comparison with the fact that SRS generation at the cavity length detuning up to +50 μm leads to the time delay of the Stokes pulses relative to the pump pulse of about 0.6ps per round trip. The time delay between (${\nu }_1+{\nu }_2$)-shifted Stokes and (${\nu }_1$)-shifted Stokes components, due to dispersion in the SrMoO₄ crystal ($n_{1.063}=1.8804$, $n_{1.173}=1.8776$ and $n_{1.220}=1.8766$ [86]), is only 0.06 ps at L = 1.7 cm per round trip. It can be comparable with the cavity length detuning time delay (0.6 ps) at 10-fold increase of the crystal length. So, in very long crystals such as the 8-cm long BaWO₄ [82], where the short-shifted (${\nu }_1+2{\nu }_2$) second Stokes component and also the long-shifted ($2{\nu }_1$) second Stokes component were not generated (Figure 3a), the dispersion effect can be significant. But, as we believe, in shorter crystals such as the used SrMoO₄ and SrWO₄ crystals, the parametric coupling effect is predominant allowing the higher order short-shifted SRS generation in the shortest 1.4-cm SrWO₄ crystal. On other hand, dispersion in the SrWO₄ crystal with 2 times increased length of 3.6 cm also suppress it.

Figure 6 shows dependences of the output SRS pulse energy on the input pump pulse energy in the Raman lasers based on (a) the 1.7-cm long SrMoO₄ crystal [88] and (b) the 1.4-cm long SrWO₄ crystal at the output coupler OC1 for the case of perfect synchronization. In both the SrMoO₄ and SrWO₄ crystals, the single ${\nu }_1$-shifted Stokes component was generated with the OC1. Slope efficiencies are comparable and amount of 17.3% and 15.2% in SrMoO₄ and SrWO₄, respectively. Output pulse energy reached 16 nJ and 13 nJ for the SrMoO₄ and SrWO₄ Raman lasers, respectively, at maximum pump pulse energy of 221 nJ. In the SrMoO₄ crystal with higher Raman gain, the SRS oscillation threshold was lower.

Let make estimations of the SRS oscillation threshold using expression (14) (we have $t_p=500\mathsf{\mu }\mathrm{s}\gg {\tau }_c=6.7\mathrm{ns}$). The output coupler OC1 has reflectivity of R_S = 87% and 89% for the SrMoO₄ and SrWO₄ Raman lasers, respectively. The round-trip transmission of the intracavity medium (intracavity losses) can be estimated as $T_S\approx 98\%$ and the cavity losses increment can be estimated as $K=ln\left(1/T_S{R}_S\right)$ ≈ 0.16 and 0.14 for these lasers, respectively. Therefore, according to expression (14) at $p=1$, we need to realize $G_{th}$ ≈ 0.16 and 0.14 for these lasers, respectively. Selecting the values of $G_0$, we find that the SRS gain increment $G$ in expression (8) is equal to the threshold value of 0.16 or 0.14 at $G_{0th}\approx$ 0.44 or 0.385 for the SrMoO₄ or SrWO₄ crystals, respectively. Numerical solution of the initial Equation (4) gives the same result. We also need to take into account strong focusing of the pump beam into the centre of the Raman-active crystal to the focal spot 35 × 30 µm² in area. An effective length of SRS interaction $L_{eff}$ for the strongly focused beam is lower than the Raman-active crystal length $L$. It can be determined [35] as

$$L_{eff}=2z_R\arctan (L/2z_R),$$

(16)

where $z_R=\pi r_p^{2}n/{\lambda }_p{M}^2$ is the Rayleigh length of a waist of the pump beam, $r_p$ is the focal spot radius of the pump beam, $n$ is the refractive index of the crystal. The effective interaction length $L_{eff}$ should be used instead of $L$ in the SRS oscillation threshold pump energy estimations. Due to Brewster incidence, we have an astigmatic beam in the crystal with the increased size of the beam in the horizontal axis (x) by a value of refractive index n. Thus, we have the focal spot radii of $r_{px}=35·n$ µm and $r_{py}=$ 30 µm in the crystal. Using the mean value of $z_R$, in expression (16), we obtain $L_{eff}$ ≈ 1.5 cm and 1.3 cm for the SrMoO₄ and SrWO₄ crystals, respectively. So, the threshold pump intensity should be about $I_{p,th}\approx G_{0th}/g{L}_{eff}\approx$ 52 MW/cm² or 63 MW/cm² corresponding to the threshold pump pulse energy of about $I_{p,th}{\tau }_p\pi r_{px}{r}_{py}\approx$ 120 nJ or 140 nJ for the SrMoO₄ or SrWO₄ Raman lasers, respectively. These calculations are in a good agreement with the experimental results (Figure 6).

Figure 7 shows dependences of output SRS pulse energy on the input pump pulse energy in the Raman lasers based on 1.7-cm long SrMoO₄ crystal [88] (Figure 7a) and 1.4-cm long SrWO₄ crystal (Figure 7b) with the output coupler OC2 for the case of perfect synchronization.

As one can see, in contrast to the previous case, the output coupler OC2 allowed to achieve more efficient SRS operation of the SrWO₄ laser (7.1%, 6 nJ) than the SrMoO₄ laser (5.4%, 5 nJ) and it is also cascade-like with combined Raman shifts for the both. Note that the SrWO₄ Raman laser generation is more efficient because of higher efficiency of the secondary SRS conversion from the ${\nu }_1$-shifted SRS component into the (${\nu }_1+{\nu }_2$)-shifted SRS component (4.5% against 3.6%). It can be explained by that the vibrational ${\nu }_2$ Raman line is more intense in comparison with the ${\nu }_1$ Raman line for SrWO₄ ($I_1/I_2=3.5$, Table 1) than for SrMoO₄ ($I_1/I_2=4.8$, Table 1). As a result, the (${\nu }_1+{\nu }_2$)-shifted SRS pulse energy (3.9 nJ at 1227 nm) in the SrWO₄ laser was 1.44 times higher than in the SrMoO₄ laser (2.7 nJ at 1220 nm). It even allowed to obtain the next cascade (${\nu }_1+2{\nu }_2$) of the short-shifted SRS conversion at 1279 nm with a pulse energy of 0.4 nJ in the SrWO₄ Raman laser (Figure 7b).

In order to achieve the strongest SRS pulse shortening, we realized the Raman laser cavity length detuning relative to the case of perfect synchronization (zero detuning). Figure 8 demonstrates the measured dependences of the output pulse energy (Figure 8a,b), the output pulse duration (Figure 8c,d) and output pulse peak power (Figure 8e,f) on the cavity length detuning for the SrMoO₄ (Figure 8a,c,e) and SrWO₄ (Figure 8b,d,f) lasers at OC2 under pumping with the maximal pulse energy of 221 nJ.

It can be seen from Figure 8a,b that SRS generation was observed only in a narrow range of the positive cavity length detuning from 0 up to +50 µm while the detuning range was essentially wider at the negative cavity length detuning (from 0 down to −220 µm). It is similar as in other synchronously pumped Raman lasers [65,66,67,68,69,70,71,72,73,74,75,76,77,78] and can be explained by non-efficient interaction between the pump and SRS pulses when only the leading edge of the SRS pulse is amplified at positive detuning. However, namely positive detuning (up to +50 µm) of the cavity length has allowed to obtain the strongest SRS pulse shortening, as can be seen from Figure 8c,d.

It is necessary to note that the (${\nu }_1+{\nu }_2$)-shifted (at 1220 or 1227 nm in the SrMoO₄ or SrWO₄ laser) and (${\nu }_1+2{\nu }_2$)-shifted (at 1279 nm in the SrWO₄ laser) Stokes pulses had strong self-shortening in the whole range of cavity length detuning and did not exceed 7 ps at negative and zero detuning of the cavity length. It had the strongest pulse shortening down to about 1.4 ps at the cavity length positive detuning of +50 μm in both Raman lasers (Figure 8c,d). The shortest pulse durations of 1.46 ps and 1.41 ps for the SrMoO₄ and SrWO₄ lasers, respectively, were close to the limit given by the inverse width of the widest ${\nu }_2$ vibrational line ($1/\pi c\Delta {\nu }_2$ = 1.0 ps and 1.2 ps for SrMoO₄ and SrWO₄, respectively). This extreme pulse shortening can be explained by the theory of ultra-short SRS pulse formation at intracavity pumping [62] taking into account that generation of the highly shortened (${\nu }_1+{\nu }_2$)-shifted or (${\nu }_1+2{\nu }_2$)-shifted SRS pulses in our extracavity Raman lasers were intracavity pumped by the ${\nu }_1$-shifted or (${\nu }_1+{\nu }_2$)-shifted SRS radiation.

The values of peak power were significantly influenced by the pulse shortening. As it can be seen in Figure 8e,f, the position of the maximum peak power at the 1220 nm for SrMoO₄ [88] and at 1227 nm for SrWO₄ has a shift from the zero detuning due to strong pulse shortening. While the output pulse energy maximum of the (${\nu }_1+{\nu }_2$)-shifted SRS component was obtained for the zero detuning (Figure 8a,b) but the peak power of its component reached the maximum value for detuning of +20 µm (Figure 8e,f) in both the lasers. For this setup, the peak power levels of 500 W and 650 W were achieved in the SrMoO₄ and SrWO₄ lasers, respectively. It is higher than the peak power of the ${\nu }_1$-shifted SRS component generated not only at the OC2 but also at the OC1 in the both Raman lasers.

6. Conclusions

In this paper, we have presented a comparative investigation of characteristics of spontaneous and stimulated Raman scattering in different alkali-earth tungstate and molybdate crystals at both high (${\nu }_1$) and low (${\nu }_2$) wavenumber anionic group vibrations. It was found that, among these crystals, SrMoO₄ and SrWO₄ crystals are the most perspective for SRS generation on both the stretching and bending modes of internal anionic group vibrations with the strongest SRS pulse shortening under synchronous laser pumping. It is caused by the fact that they have not only highly intense stretching mode ${\nu }_1$ Raman line for efficient primary extracavity long-shifted SRS conversion but also the widest bending mode ${\nu }_2$ Raman line for the strongest SRS pulse shortening down to the inverse width of the widest Raman line at the secondary intracavity short-shifted SRS conversion. The possibilities of the strongest 26-fold pulse shortening down to 1.4 ps at the Stokes component with the combined (${\nu }_1+{\nu }_2$) Raman shift of the synchronously pumped extracavity Raman lasers based on SrMoO₄ and SrWO₄ crystals have been demonstrated. This value of 1.4 ps is close to the inverse width of the widest ${\nu }_2$ Raman line ($1/\pi c\Delta {\nu }_2\approx$ 1 ps). It was found that synchronously pumped cascade SRS with combined (${\nu }_1+{\nu }_2$) Raman shift is more efficient in the SrWO₄ crystal because the vibrational ${\nu }_2$ Raman line is more intensive in comparison with the vibrational ${\nu }_1$ Raman line for SrWO₄ than for SrMoO₄. It was also found that the high order Stokes and anti-Stokes short-shifted SRS generation under extracavity synchronous pumping can be reached using the crystals with a short length (~1 cm) in comparison with a coherence length of parametric Raman interaction of the long-shifted and short-shifted SRS components but in long crystals (>3 cm), the higher-order short-shifted and long-shifted SRS generation is suppressed due to a dispersion effect.