**6. Features of ytterbium lasing: line spectra, data interpretation**

Line spectra observed in the crystal and glass relate to the second stage of Yb-lasing. The second stage is the Yb-lasing in the resonator after the end of pump due to inversion remained in the medium. Of special interest the details of Yb spectra are. Twisted, inclined and structured lines in spectra were observed in the region 1.03–1.05 μm, Figs.11-14,16. Figures 11, 13, 14 showed an inclination of some lines to the short-wavelength side of the spectrum in respect to the vertical position of the slit. Nearly all line spectra showed quasiperiodic structures of small-scale (50–200 μm) spots (Figs. 11, 14, 16). Twisted lines consisting of separate spots are shown at Fig.16. This picture indicates a wave-like change in the generation wavelength in separate lasing spots along the height of the slit.

Fig. 16. Twisted lines with multiple spots of Yb:glass generation at several transitions near 1.04 μm; time delay from the pump pulse ≈ 50 ns

`instant photograph' of the spatial arrangement of such a structure. The photons with the wavelength resonant with the structure, *λg/ng = mΛ*, will occupy different spatial positions in the `photograph' with respect to the layers of thickness *linv « λg/ng*. And only a small part of the total number of photons in the `resonator' of our DFB laser will coincide in position and phase with the inversion maxima in the medium and, therefore, will be amplified efficiently during the lifetime *tinv < T* of the layers of the given spatial configuration, forming a beam on the basis of photon trains. During the movement of the inversion layers structure (following the movement of the hypersonic wave), new groups of phased photons will be formed, and their frequency will shift with changing the structure parameters. In such a way the model considered above explains the appearance of the sharp angular radiation pattern of the broadband lasing of ytterbium in the crystal and glass. In the case of a `thick' sinusoidal grating, photons occupying different positions in space (which is equivalent to the large phase dispersion within the photon train) are amplified. This leads to the increase in the radiation divergence. Thus, when a periodic spatial grating of thin inversion layers is produced in the active medium, the high-directional stimulated radiation of an ensemble of

excited atoms can be observed in the optical range.

1.04 μm; time delay from the pump pulse ≈ 50 ns

**6. Features of ytterbium lasing: line spectra, data interpretation** 

the generation wavelength in separate lasing spots along the height of the slit.

Line spectra observed in the crystal and glass relate to the second stage of Yb-lasing. The second stage is the Yb-lasing in the resonator after the end of pump due to inversion remained in the medium. Of special interest the details of Yb spectra are. Twisted, inclined and structured lines in spectra were observed in the region 1.03–1.05 μm, Figs.11-14,16. Figures 11, 13, 14 showed an inclination of some lines to the short-wavelength side of the spectrum in respect to the vertical position of the slit. Nearly all line spectra showed quasiperiodic structures of small-scale (50–200 μm) spots (Figs. 11, 14, 16). Twisted lines consisting of separate spots are shown at Fig.16. This picture indicates a wave-like change in

Fig. 16. Twisted lines with multiple spots of Yb:glass generation at several transitions near

Consider the conditions for the development of lasing in a resonator at the end of pump, under disappearance of longitudinal hypersonic waves and relaxation of the excitation region in the active medium. Lasing developed in the resonator in the presence of the region of strong optical inhomogeneity near the active element surface- a thin layer with the inhomogeneous inversion distribution and gradient of the refractive index, *n* produced by pressure, temperature, and density gradients. To the rest part of the sample the pump penetrated weakly. The refractive index profile was formed during the action of the pump on a medium and it also changed after the end of pump during the relaxation of the excited region. This profile affected the development of Yb-lasing. An experimental study of a nonstationary refractive index profile (so-called "transient lens") arising under short laser pulse interaction with a medium is a rather complicated task. Here we discuss in a qualitative form processes that affected formation of the index profile, *n*(*r*) assuming that pumping was symmetrical over the azimuth angle. Consider again a part of the medium in the form of a cylinder of 250 *μ*m diameter and 100 *μ*m length, which fits the size and configuration of the focal region at tight ccl focusing. The heat load occurred in such a cylinder during inversion formation, Yb-lasing, and mainly, due to pump radiation scattering. The heat load was estimated to be <10 mJ, which allows for the average thermal energy density in the cylinder, *<* 2 *×* 103 J/cm3. Estimations for the average pressure, *P*av and temperature, *T*av just after the end of pump give *P*av *≈* 1 GPa and *T*av *≈*103 K. In actuality, due to the bell-shaped profile of the pump beam, *P*(*r*) and *T*(*r*) profiles with maxima exceeding averaged values arose in the medium. The maximum pressure near the axis of the excitation region at the moment near the end of pump pulse may be estimated by using the value *P*pt, at which a phase transition occurs. Structure modifications similar to phase transitions had been observed in Yb:YAG and Yb:glass samples near the axis, where hypersonic waves of maximum intensity propagated (section 3). We use data for glass *P*pt *≈* 10 GPa, *δn*/*δP* =-0.13 *×* 10–5 bar–1, and *δn*/*δT* = –6 *×* 10–6 K–1 (Alcock & Emmony, 2002; Koechner, 2006; Mak et al., 1990). At *P*(0) *≈ P*pt the drop of *n*(*r*) along *r* from axis to periphery of the cylinder due to the drop in pressure in the medium is *Δn*(*P*pt) = *δn*/*δP ×P*pt = 0.13. Changes of *Δn* due to temperature gradients are significantly less. For example, at *ΔT* = 103 K, *Δn*(*T*) = *δn*/*δT × T* = –6 *×* 10–3. For YAG *δn*/*δT ≈* 8 *×*10–6 K–1 (Koechner, 2006; Mak et al., 1990) and *Δn*(*T*) *≈* 8 *×* 10–3 .Thus, the main contribution to the *n*(*r*) profile was made just by pressure. The *n*(*r*) profile defines a sign, optical power and aberrations of a "transient" lens in the medium. Optical power of the lens after the end of pump can be estimated at the assumption that the index had been changed by *Δn ≈* 0.1 at *Δr ≈* 150 *μ*m. That gradient corresponds to appearance in the medium of a positive lens with diameter 2*r ≈* 300 *μ*m and *f ≈* 1cm. A simplified scheme for the case of lasing in the resonator with a "transient" lens is given in Fig. 17.

The evolution of profiles *P*(*r*) and *n*(*r*) after pumping was determined by unloading of the high-pressure region in the focal volume. A problem of unloading of a small cylindrical or spherical region in a solid-state optical medium, where the energy had been stored after ns laser pulse irradiation, had been considered in several studies (Bullough & Gilman, 1966; Conners & Thompson, 1966; Sharma & Rieckhoff, 1970). A typical process of relaxation is the propagation of an elastic dilatational wave across the medium. The time of traveling, *ttr* of an elastic wave with the sound velocity across the excitation region can be evaluated by *ttr* =*r*/*vs* . For the glass, at *rp* = 250 μm and *vs* = 4.5 ×105 cm/s, *ttr* ≈ 50 ns. The propagation of the dilatational wave along radial direction outward the center of the pumped region had led to the pressure profile deformation. With the drop of pressure at the axis, the gradient *dn*/*dr* 

Excitation of Periodical Shock Waves in Solid–State Optical Media (Yb:YAG, Glass)

spots, which dimensions characterized beam divergence in each of the channels.

correspond sources of Yb emission with dimensions of several wavelengths only.

Structures of multiple spots in Yb:YAG and Yb:glass lines (Figs. 11, 14, 16) may find explanation when one takes into consideration the specific spatial configuration of the field of thermo-elastic stresses in the area of ns laser pulse focusing into the medium. The quasiperiodic, alternating in sign, oscillating character of the amplitude of the tangential component of stress in dependence on the radial coordinate inside a small spherical region of a solid-state optical medium at ns laser pulse focusing was ascertained (Conners & Thompson, 1966; Sharma & Rieckhoff, 1970). In spherical ring zones round the centre of the focal region the tangential tensile stresses in the medium are consequently replaced by compressing stresses, and then, again, by the tensile ones, etc. The spatial period of such oscillations calculated for the case of ≈70 mJ laser pulse focusing into the glass constituted 30 μm (Sharma & Rieckhoff, 1970). In conditions of our experiment one can expect the occurrence of oscillating (in space and time) profiles of thermo-elastic stresses in Yb-doped media within the area of ccl radiation focusing as well. Alternating in sign stresses should result in a small-scale modulation at the profile *n*(*r*), Fig.17. The oscillation of the *n*(*r*) profile should stimulate the Yb generation in ring zones, which may fall into separate generation channels. The structures of bends consisting of generation spots, Figs.16 may be considered as a kind of an "image" of the distribution over the radial coordinate of tangential stress peaks in the focal region projected with a magnification on the slit. It is possible to say that such images were taken by the high-speed photography method. The "illumination" for this high-speed photography came from the Yb laser pulse itself. The "exposure" time of a single frame corresponded to the duration of the generation pulse. Time delay between the pump pulse and the "shooting" moments makes ≈ 50ns (Figs. 16) and over 300 ns (Fig.14). Splitting of a single bend into several generation spots with wavelength shifts (Fig.16) is in agreement with the model of generation channels. The shift of a spot to the long-wavelength side corresponds to increase of pressure (stresses) in the medium, and the shift to the short-

at SBS of Focused Low–Coherent Pump Radiation: Structure Changes, Features of Lasing 391

same phase made several round trips in the resonator during the time *t*', and a beam of the high directivity with the generation wavelength λg corresponding to one of the longitudinal modes was formed. So, generation in different channels over the pumped area developed at their own frequencies, and this radiation, reaching the slit at different points gave us an integrated picture of lasing beams and the spectrum in the form of lines composed of small

For Yb-doped plates placed in the middle of the resonator with *L* = 20 mm and for trajectories located in plates at *r* < 500 μm, angles *δ* should lie inside the spectrograph registration angle ≈10–2 rad. Beams of generation left resonator at angles *δ* and reached the slit. The greater was the distance *r* between the generation trajectory inside the sample and the resonator axis, the greater was the shift of the corresponding generation beam spot upward in vertical direction on the slit, Fig. 17. So, under the propagation of beams of highdirectivity from resonator at angles δ, one, actually, observed at the slit a projection (a magnified "image") of a spatial distribution of generation channels in the active medium over the radial coordinate. Since data on the Yb emission intensity distribution over the azimuth angle were absent, one can speak only about partial mapping on the slit of the spatial distribution of generation channels in the lasing area. The magnification coefficient *k*  of such an image may be estimated from comparison of dimensions of the possible generation area (*r* ≤ 500 μm) and the height of line spectra at the slit (≈1 cm), *k* ≥ 20. With account of *k* factor, to observed structures of small-scale spots in line spectra should

decreased accordingly. So, in the course of relaxation there occurred smoothing of profiles *P*(*r*) and *n*(*r*). It should be noted that profile *n*(*r*) could include a small-scale modulation as well (imitated at Fig.17). The sources for such modulation might be pressure and density perturbations that had been arising in a medium at propagation of hypersonic waves during and after pumping. Note, that typical time of thermal relaxation of the medium in a cylindrical region with radius *rp* is *τ* = *rp* 2/4μ » *t*tr. Here *rp* is the radius of the pumped region and μ is the coefficient of the temperature diffusivity. For Yb:YAG at *rp* = 250 μm, μ = 0.046 cm2s–1 (Koechner, 2006 ), *τ* ≈ 3 ms. Hence, the time of thermal relaxation of medium is several orders higher than dynamic unloading. That is, roughly, a picture of formation and evolution of the refractive index profile in the medium in the region of ccl focusing defined by a non-uniform heat load and dynamics of the pressure profile during pumping and under the medium relaxation. So, a sequence of acoustic waves diverging in radial direction outward the focal region should affect the build-up of Yb-lasing in the resonator.

Fig. 17. Scheme of the experiment on excitation of Yb-doped medium in the resonator by focused ccl radiation: 1 - active element; 2, 3 - mirrors; 4 - spectrograph STE-1 located at distance S≈1m from the resonator; 5 - a picture imitating spectra registered; δ –angle between a generation trajectory and resonator axis. The dimensions of the resonator with active element are exaggerated relative to the scale of the scheme. Profiles of pump beam intensity, *I(r)* and pressure in the medium, *P(r)* are out of scale as well

The interpretation of data on the Yb-lasing in the resonator has required also the introduction of a new concept for the photon electromagnetic field distribution in space, considered in section 5. This new approach allows the existence of light beams of an aperture *d* ≈ *λ* and with a "sub-diffraction" angle of divergence. Within the framework of this new concept the picture of Yb lasing in the resonator looks as follows. The location of line spectra in the upper part of the slit evidences that Yb radiation emergent from the resonator was deflected from the axis, i.e., off-axis oscillations were built up in the resonator. For the arrangement of mirrors and active element given in our experiment off-axis lasing could be developed over trajectories with reflections at angles *δ*1, *δ*2 … at the mirror (*3*), Fig. 17. Such trajectories located at distances *r*pt < *r* < *rp* from the axis, should contain curvilinear segments in the area of the optical inhomogeneity. We suppose that beams with aperture of several *λ* (small generation channels) constituted lasing along these off-axis trajectories. At the high value of *dn*/*dr* and the very high level of inversion, in each of the channels the selection of emitted photons (in accordance with their phases) took place in such a way that only a group of photons being approximately in the same phase φ0 overcame the generation threshold. The photons different by phase from φ0 were scattered due to diffraction and left the channel aperture. In the given channel with a coordinate *r* a group of photons of the

decreased accordingly. So, in the course of relaxation there occurred smoothing of profiles *P*(*r*) and *n*(*r*). It should be noted that profile *n*(*r*) could include a small-scale modulation as well (imitated at Fig.17). The sources for such modulation might be pressure and density perturbations that had been arising in a medium at propagation of hypersonic waves during and after pumping. Note, that typical time of thermal relaxation of the medium in a

and μ is the coefficient of the temperature diffusivity. For Yb:YAG at *rp* = 250 μm, μ = 0.046 cm2s–1 (Koechner, 2006 ), *τ* ≈ 3 ms. Hence, the time of thermal relaxation of medium is several orders higher than dynamic unloading. That is, roughly, a picture of formation and evolution of the refractive index profile in the medium in the region of ccl focusing defined by a non-uniform heat load and dynamics of the pressure profile during pumping and under the medium relaxation. So, a sequence of acoustic waves diverging in radial direction

outward the focal region should affect the build-up of Yb-lasing in the resonator.

Fig. 17. Scheme of the experiment on excitation of Yb-doped medium in the resonator by focused ccl radiation: 1 - active element; 2, 3 - mirrors; 4 - spectrograph STE-1 located at distance S≈1m from the resonator; 5 - a picture imitating spectra registered; δ –angle between a generation trajectory and resonator axis. The dimensions of the resonator with active element are exaggerated relative to the scale of the scheme. Profiles of pump beam

The interpretation of data on the Yb-lasing in the resonator has required also the introduction of a new concept for the photon electromagnetic field distribution in space, considered in section 5. This new approach allows the existence of light beams of an aperture *d* ≈ *λ* and with a "sub-diffraction" angle of divergence. Within the framework of this new concept the picture of Yb lasing in the resonator looks as follows. The location of line spectra in the upper part of the slit evidences that Yb radiation emergent from the resonator was deflected from the axis, i.e., off-axis oscillations were built up in the resonator. For the arrangement of mirrors and active element given in our experiment off-axis lasing could be developed over trajectories with reflections at angles *δ*1, *δ*2 … at the mirror (*3*), Fig. 17. Such trajectories located at distances *r*pt < *r* < *rp* from the axis, should contain curvilinear segments in the area of the optical inhomogeneity. We suppose that beams with aperture of several *λ* (small generation channels) constituted lasing along these off-axis trajectories. At the high value of *dn*/*dr* and the very high level of inversion, in each of the channels the selection of emitted photons (in accordance with their phases) took place in such a way that only a group of photons being approximately in the same phase φ0 overcame the generation threshold. The photons different by phase from φ0 were scattered due to diffraction and left the channel aperture. In the given channel with a coordinate *r* a group of photons of the

intensity, *I(r)* and pressure in the medium, *P(r)* are out of scale as well

2/4μ » *t*tr. Here *rp* is the radius of the pumped region

cylindrical region with radius *rp* is *τ* = *rp*

same phase made several round trips in the resonator during the time *t*', and a beam of the high directivity with the generation wavelength λg corresponding to one of the longitudinal modes was formed. So, generation in different channels over the pumped area developed at their own frequencies, and this radiation, reaching the slit at different points gave us an integrated picture of lasing beams and the spectrum in the form of lines composed of small spots, which dimensions characterized beam divergence in each of the channels.

For Yb-doped plates placed in the middle of the resonator with *L* = 20 mm and for trajectories located in plates at *r* < 500 μm, angles *δ* should lie inside the spectrograph registration angle ≈10–2 rad. Beams of generation left resonator at angles *δ* and reached the slit. The greater was the distance *r* between the generation trajectory inside the sample and the resonator axis, the greater was the shift of the corresponding generation beam spot upward in vertical direction on the slit, Fig. 17. So, under the propagation of beams of highdirectivity from resonator at angles δ, one, actually, observed at the slit a projection (a magnified "image") of a spatial distribution of generation channels in the active medium over the radial coordinate. Since data on the Yb emission intensity distribution over the azimuth angle were absent, one can speak only about partial mapping on the slit of the spatial distribution of generation channels in the lasing area. The magnification coefficient *k*  of such an image may be estimated from comparison of dimensions of the possible generation area (*r* ≤ 500 μm) and the height of line spectra at the slit (≈1 cm), *k* ≥ 20. With account of *k* factor, to observed structures of small-scale spots in line spectra should correspond sources of Yb emission with dimensions of several wavelengths only.

Structures of multiple spots in Yb:YAG and Yb:glass lines (Figs. 11, 14, 16) may find explanation when one takes into consideration the specific spatial configuration of the field of thermo-elastic stresses in the area of ns laser pulse focusing into the medium. The quasiperiodic, alternating in sign, oscillating character of the amplitude of the tangential component of stress in dependence on the radial coordinate inside a small spherical region of a solid-state optical medium at ns laser pulse focusing was ascertained (Conners & Thompson, 1966; Sharma & Rieckhoff, 1970). In spherical ring zones round the centre of the focal region the tangential tensile stresses in the medium are consequently replaced by compressing stresses, and then, again, by the tensile ones, etc. The spatial period of such oscillations calculated for the case of ≈70 mJ laser pulse focusing into the glass constituted 30 μm (Sharma & Rieckhoff, 1970). In conditions of our experiment one can expect the occurrence of oscillating (in space and time) profiles of thermo-elastic stresses in Yb-doped media within the area of ccl radiation focusing as well. Alternating in sign stresses should result in a small-scale modulation at the profile *n*(*r*), Fig.17. The oscillation of the *n*(*r*) profile should stimulate the Yb generation in ring zones, which may fall into separate generation channels. The structures of bends consisting of generation spots, Figs.16 may be considered as a kind of an "image" of the distribution over the radial coordinate of tangential stress peaks in the focal region projected with a magnification on the slit. It is possible to say that such images were taken by the high-speed photography method. The "illumination" for this high-speed photography came from the Yb laser pulse itself. The "exposure" time of a single frame corresponded to the duration of the generation pulse. Time delay between the pump pulse and the "shooting" moments makes ≈ 50ns (Figs. 16) and over 300 ns (Fig.14). Splitting of a single bend into several generation spots with wavelength shifts (Fig.16) is in agreement with the model of generation channels. The shift of a spot to the long-wavelength side corresponds to increase of pressure (stresses) in the medium, and the shift to the short-

Excitation of Periodical Shock Waves in Solid–State Optical Media (Yb:YAG, Glass)

at SBS of Focused Low–Coherent Pump Radiation: Structure Changes, Features of Lasing 393

the value of *dn/dP* for glass (Alcock & Emmony, 2002), we obtain *ΔP* ≈ 30 GPa. So, this estimates the pressure, which arises at the axis of the focal region after the end of pump. Similar estimations are possible for the pressure jumps, which arise during the propagation of dilatational waves outward the center of the focal region. Knowing a value of the wavelength shift in a bend at Fig.16, *Δλq (r),* one can estimate changes of the index, Δ*n* and pressure, Δ*P* in the medium, using the expression (4). As seen from Fig.16, for glass at Δλ*<sup>q</sup>* (*r*) ≈ 0.3x10–7 cm, *λ<sup>q</sup>* ≈ 10–4 cm, *L* ≈ 2 cm, *n*<sup>0</sup> ≈ 1, *Δl* ≈ 10-2 cm, Δ*n* ≈ 0.05, and Δ*P* ≈ 5 GPa. The estimated Δ*P* considerably exceeds the glass fracture strength < 0.1 GPa (Sharma & Rieckhoff, 1970), which was measured usually for applied static load. The role of tensile stresses in laser damage of transparent dielectrics was discussed in many publications (Koldunov et al., 2002; Sharma & Rieckhoff, 1970; Strekalov, 2000). It was considered that due to oscillating character of the stress amplitude in the focal area there must be observed laser damage of the medium in the form of periodically spaced spherical rings (Sharma & Rieckhoff, 1970; Strekalov, 2000). It is known that material strength sharply grows under pulsed load as compared to the static load. Under the high-speed deformation (in the ns range), strength of material becomes comparable to the theoretical limit-tens GPa (Kanel et al., 2007). So a fast periodic change of the stress sign in the medium should decelerate the development of material destruction in the form of rings. This is the reason why the laser damage in the form of multiple rings usually was not observed in many experiments when ns pulses of laser radiation were focused into the volume of transparent dielectrics. The damage in the form of rings was not observed in our experiments as well. One of the few observations of the multiple ring damage (Martinelli, 1966) relates to the case when glass samples were exposed to focused free-running laser radiation. Anyway, according to calculations (Sharma & Rieckhoff, 1970) oscillating profile of thermo-elastic stresses should occur in the region of ns laser pulse focusing. The presented material provides experimental data which confirm in the quality form the calculated (Sharma & Rieckhoff, 1970) picture of

thermo-elastic stresses distribution and elastic wave propagation across the medium.

structural changes based on the action of intense hypersonic waves were considered.

The study of the interaction of powerful ns pulses of low-coherence radiation of the LiF: F2+ color center laser (ccl) with optical materials (Yb:YAG, glass, et al.) was carried out. Efficient SBS of low-coherence pump, accompanied by SRS and formation of hypersonic waves reaching the intensity of shock waves were found. A physical model of excitation of SBS and hypersonic waves at scattering of ultrashort pulses of low-coherence pump at stationary inhomogeneities in optical materials is presented. It is shown that ns laser pulse, whose duration is much higher than its inverse spectral width, causes SBS much more efficient than a pulse of high coherence with the same duration and energy. Unlike SBS of a coherent radiation caused by a pressure fluctuation, scattering of low-coherence pump may be caused by any stationary inhomogeneities in a medium: cracks, dislocations, microinclusions, or just by a plane back surface of a sample. An effective energy contribution of light pulses into hypersonic waves on a small coherence length near the input surface of a sample leads to their transformation into a periodic succession of high-pressure shock waves, which results in structure changes of a crystal lattice (phase transition) in that region. The appearance of structural changes in optical materials that are specific to the interaction of powerful pulses of low coherence radiation with matter was found. The mechanisms of

**7. Conclusion** 

wavelength—to decrease of pressure. In the course of medium relaxation, stress amplitudes and, correspondingly, spot wavelength shifts should have been reduced. At ≈50ns delay (twisted lines, Fig.16) spots have noticeable wavelength shifts which correspond to high stress amplitudes. Structures at moments over 300 ns after pump (Fig.14) correspond to a smoothed picture of *n*(*r*) profile. Note, that structures recorded with such big delays indicate the continued acoustic "ringing" in the medium. Estimated from Fig. 11, 14, 16 (at *k* = 20) period of stress spatial oscillations in YAG and glass varies from 15 to 40 μm and is in a qualitative agreement with the calculations (Sharma & Rieckhoff, 1970). Acoustic vibrations frequencies corresponding to these values constitute 108–109 Hz. Attenuation of phonons at these frequencies at a room temperature for YAG is smaller than 0.1 db/μs and for glass about 10 cm–1 (Dutoit, 1974; Zhu et al., 1991). These data confirm that "ringing" of the unloading medium in the focal region in YAG and glass may continue over several μs.

The Yb:YAG line spectra near 1,03 μm were registered usually with 10÷50 ns delays after the end of ccl pulse and even together with the trailing edge of the ccl pulse. Observations of line spectra near 1,03μm, emitted soon after the pump pulse, like Fig.11, reveal noticeable inclination of spectral lines. This bending means that the wavelength of lasing at the same longitudinal mode (the number of nodes is preserved) changes from the centre to periphery of the excited region in the active medium. This corresponds to the development of lasing in some sites of the medium with a pressure (refractive index) gradient from the beam centre to its periphery. It can be easily shown that *λg* shifts to the blue if the refractive index gradient decreases from the beam axis to its periphery, Fig. 11. The pressure drop *ΔP* in the medium after the end of pump can be estimated from this line spectral shift. The frequency ω*<sup>q</sup>* of the longitudinal mode of the resonator with the number of wavelengths *λq* over the resonator length *2L* equal to *q* is described by the expression

$$\alpha\_q = \frac{\pi c q}{(L - \Delta l)n\_o + \Delta l \cdot n(r)}\tag{2}$$

where *Δl* is the longitudinal size of the optical inhomogeneity of radius *r*; n0 is the averaged refractive index outside the nonlinearity region; and *n(r)* is the refractive index in the nonlinear region. It follows from (2) that the change in the mode frequency ω*<sup>q</sup>* during the displacement along the radius from *r1* to *r2* is

$$
\Delta \phi\_q = \frac{\pi c q \Delta [n(r\_1) - n(r\_2)]}{[(L - \Delta l)n\_0 + \Delta l \cdot n(r\_1)][(L - \Delta l)n\_0 + \Delta l \cdot n(r\_2)]} \tag{3}
$$

By substituting the expression for 0 2[( ) ( )] ( ) *<sup>q</sup> L ln l nr q* λ *r* −Δ +Δ ⋅ <sup>=</sup> into (3) and assuming that

*Δln(r)<Ln0*, we obtain the dependence of the change in the refractive index, *Δn* on *r*

$$
\Delta \eta = \frac{\Delta \vec{\lambda}\_q(r) L n\_0}{\vec{\lambda}\_q \Delta l} \tag{4}
$$

Here, *Δλq(r)* is the wavelength shift along the radius. For *Δλq(r)* ≈ 1.4 ×10-7 cm, *λ<sup>q</sup>* ≈ 10-4 cm, L ≈ 2 cm, *n0* ≈ 1, and *Δl* ≈ 10-2 cm, the change in the index is *Δn* ≈ 0.28. By assuming that the change in the index is produced only by the change in pressure along the radius and using the value of *dn/dP* for glass (Alcock & Emmony, 2002), we obtain *ΔP* ≈ 30 GPa. So, this estimates the pressure, which arises at the axis of the focal region after the end of pump.

Similar estimations are possible for the pressure jumps, which arise during the propagation of dilatational waves outward the center of the focal region. Knowing a value of the wavelength shift in a bend at Fig.16, *Δλq (r),* one can estimate changes of the index, Δ*n* and pressure, Δ*P* in the medium, using the expression (4). As seen from Fig.16, for glass at Δλ*<sup>q</sup>* (*r*) ≈ 0.3x10–7 cm, *λ<sup>q</sup>* ≈ 10–4 cm, *L* ≈ 2 cm, *n*<sup>0</sup> ≈ 1, *Δl* ≈ 10-2 cm, Δ*n* ≈ 0.05, and Δ*P* ≈ 5 GPa. The estimated Δ*P* considerably exceeds the glass fracture strength < 0.1 GPa (Sharma & Rieckhoff, 1970), which was measured usually for applied static load. The role of tensile stresses in laser damage of transparent dielectrics was discussed in many publications (Koldunov et al., 2002; Sharma & Rieckhoff, 1970; Strekalov, 2000). It was considered that due to oscillating character of the stress amplitude in the focal area there must be observed laser damage of the medium in the form of periodically spaced spherical rings (Sharma & Rieckhoff, 1970; Strekalov, 2000). It is known that material strength sharply grows under pulsed load as compared to the static load. Under the high-speed deformation (in the ns range), strength of material becomes comparable to the theoretical limit-tens GPa (Kanel et al., 2007). So a fast periodic change of the stress sign in the medium should decelerate the development of material destruction in the form of rings. This is the reason why the laser damage in the form of multiple rings usually was not observed in many experiments when ns pulses of laser radiation were focused into the volume of transparent dielectrics. The damage in the form of rings was not observed in our experiments as well. One of the few observations of the multiple ring damage (Martinelli, 1966) relates to the case when glass samples were exposed to focused free-running laser radiation. Anyway, according to calculations (Sharma & Rieckhoff, 1970) oscillating profile of thermo-elastic stresses should occur in the region of ns laser pulse focusing. The presented material provides experimental data which confirm in the quality form the calculated (Sharma & Rieckhoff, 1970) picture of thermo-elastic stresses distribution and elastic wave propagation across the medium.
