Файл: Antsiferov V.V., Smirnov G.I. Physics of solid-state lasers (ISBN 1898326177) (CISP, 2005)(179s).pdf

Physics of Solid-State Lasers

7. The most efficient method of smoothing the spatial heterogeneity of the field in the active medium of the solid-state laser with flat mirrors is the method of compensated phase modulation (CPM) [1,31]. The CPM method and its advantages are based on the fact that with nonstationary elements of the resonator and the constant length of the resonator, the pattern of the field of standing waves is displaced in relation to the active medium by means of two phase modulators (2 and 6 in Fig. 1.1) positioned on both sides of the active rod; the modulators receive the sinusoidal voltage in the opposite phase from the generator 15. The modulation frequency should satisfy the relationship ω m ≥ 2π /ts, ts is the duration of the lasing spike. The phase modulators have the form of electro-optic crystals (KDP, LiNbO3, etc.), whose mutual orientation is regulated by means of two crossed polarisers and the linearly polarised radiation of the tuning laser.

To develop the homogeneous field in the active medium between the phase modulators, the intensity of the standing waves

must be equal to I0/2 at any point x. The following condition must be satisfied in this case:

The first integral is equal to 0 because it is antisymmetric with respect to t, and the second integral is expressed by means of Bessel’s function. Consequently, the criterion (1.3) is reduced to the condition J0(2β ) = 0, where J0 is the Bessel function of the zero order. The first zero of this function is situated at the point β = 0.38π . If the phase modulators are represented by the KDP crystals, to develop a homogeneous field between them, the strength of the modulated electric field on the crystals must be Ez = βλ /π ln0r63. At the length of the crystals l = 5 cm, n0 = 1.5, r63 = 8.5 × 10 –10 cm/V, the strength of the electric field is Ez = 1.8 kV/cm, which is in good agreement with the experimental results.

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Solid-state chromium lasers in free lasing regime

1.2.4 The methods of selection wavelength tuning of the radiation wave

Extensive investigations carried out using single-frequency lasers have stimulated the development of effective methods of controlling the spectral characteristics of laser radiation. The operating principle of the mechanism of selection of longitudinal modes is based on the addition into the laser resonator of the selective dependence of the losses on the frequency of radiation using dispersion elements: prisms, diffraction gratings, interference–polarisation filters, interferometers, resonance reflectors etc. The effect of all dispersion resonators is based on the two mechanisms of selection of the longitudinal modes: angular and amplitude.

In the resonators with the angular mechanism of selection, the bundles of radiation of different longitudinal modes propagate under different angles in relation to the axis of the resonator and the losses differ. The losses will be minimum in the mode for which the resonator was tuned. In the case of amplitude selection, the resonator is tuned for longitudinal modes and the difference in the losses is caused by the spectral dependence of effective transparency, reflectance or polarisation characteristics of the dispersion element.

In the dispersion resonator, the half width of the curve of the dependence of the losses on frequency γ (ω i) is considerably smaller than the half width of the gain line. The threshold value of amplification is obtained at the point of contact of these two curves, and the wavelength of laser radiation is changed by displacing the curve of the selective losses in relation to the gain line. In practice, this is carried out in most cases using combined dispersion resonators, containing several dispersion elements, one of which ensures a narrow lasing spectrum and the other ones a wide range of radiation wavelength tuning.

The intensity of laser radiation I(ω i) at the frequency ω i for a single beat of the resonator changes in the following manner: Ik+1(ω i)/

Ik(ω i) = exp δ (ω i), where δ (ω i) = α (ω i) – γ (ω i), α (ω i) is the amplification coefficient at frequency ω i. At the increment δ > 0, the intensity of

radiation increases, and at δ < 0 it decreases. The difference in the coefficient of the losses, required for suppressing weaker modes, is inversely proportional to the number of passages k of the photons through the resonator during which the intensity reaches the maximum level from the level of spontaneous noise

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In the free lasing regime k ~ 103 and the required difference of the losses for the two modes at which the intensities differ by an order of magnitude at the end of the linear development of lasing is very small: (γ 2 – γ 1) ~ 10–3. In the quasistationary regime, relatively weak mode discrimination is sufficient to ensure lasing on a single longitudinal mode by means of conventional dispersion elements.

1.3 RUBY LASERS

The lasing of forced radiation in the optical region of the spectrum was obtained for the first time in a ruby laser (Cr3+:Al2O3). Ruby crystals are characterised by high strength, homogeneity and heat conductivity. The optimum concentration of the impurity ions of Cr is approximately 0.05%, which corresponds to the density of the active centres of 1.6 × 10 19 cm–3, and at this concentration the ruby is rose-coloured. Ruby is an anisotropic single crystal with a trigonal symmetry in which the trigonal axis coincides with the optical axis of the crystal. The refractive index of ruby at the lasing wavelength is: no = 1.763, ne = 1.755. The melting point of the ruby crystal is 2040°C, the hardness according to Moose is 9, density 3.98 g/

Fig. 1.3. Energy levels E (eV) of chromium ions Cr3+ in sapphire (ruby).

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Solid-state chromium lasers in free lasing regime

cm3, the thermal expansion coefficient 5.31 (4.78)×10 –6 deg–1, the thermal optical constant dn/dT = 1.0×10 –6 deg–1. The ruby crystal is characterised by the highest heat conductivity (42 W/m grad) of all other active media.

The diagram of the levels of the Cr3+ ion in the corundum crystal is shown in Fig. 1.3 in the most widely used symbols. The gain line of ruby is uniformly broadened, and the half width of the R1- line at room temperature is 0.53 nm and at the liquid nitrogen temperature it decreases to 0.014 nm. During heating of ruby, the centre of the gain line is displaced to the wavelength range of the spectrum with a speed of 6.3 × 10 –3 nm/deg. The excitation of the metastable level

the width of each band of 100 nm. These absorption bands are connected by rapid (~100 ns) radiationless transitions with the metastable level 2E. Thus, the ruby laser operates by the three-level mechanism.

1.3.1 Spectral–time characteristics of free lasing TEMooq in a laser with flat mirrors

The spectral–time and energy parameters of a ruby laser in the free lasing regime have been investigated and analysed by the authors of this book in a number of investigations.

The free lasing of longitudinal (TEMooq) modes in a ruby laser with flat mirrors always took place in the regime of non-attenuat- ing pulsations of radiation intensity, Fig.1.4. In the case of the nonoptimum parameters of the flat resonator there were periodic jumps of the radiation wavelength to the short wave length region of the spectrum, and the radiation pulsations were completely irregular as regards the amplitude and repetition frequency, Fig 1.4c. After optimisation of the resonator parameters (length 1.6 m, the diameter of the diaphragms d = 1.4 mm) and the dynamic tuning of the elements of the resonator using the method described previously, the pulsations of the intensity of radiation became regular as regards the repetition frequency and there was an unidirectional displacement of the wavelength of lasing throughout the entire pumping pulse, Fig. 1.4d.

The residual non-regularity of the radiation pulsations with respect to intensity is associated with the effect of oscillations of the resonator mirrors. In the first lasing peak it was possible to excite a large number of longitudinal modes in the band of the spectrum with a width of approximately 25 pm. During 3–4 peaks of lasing, the number of excited modes was greatly reduced to a single longitudinal mode. This was caused by the effect of a strong dispersion

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Physics of Solid-State Lasers

Fig. 1.4. Parameters of lasing of TEMooq modes of a ruby laser (RL2B 8×120/180 mm) with flat mirrors in normal conditions (c) and with the effect of technical perturbations of the resonator removed (d,e): a) oscillation of radiation intensity, 20 µs markers; b) time evolvement of the distribution of radiation intensity in the long-range zone; c–e) time evolvement of the lasing spectrum without selection of longitudinal modes (c,d) and with selection (e), regions of dispersion of interferometers 24 pm (c,d) and 8 pm (e). Ep = 3Et, length of the resonator L 1 m (c) and 1.6 m (d,e), diameters of diaphragms 1.0 mm (c) and 1.4 mm (a,b,d,e).

of the amplification coefficients of different modes determined by the formation of a longitudinal heterogeneity of inversion in the active medium. In subsequent stages, a single longitudinal mode was excited in every successive peak, and the change of the modes from peak to peak took place in the jumps whose spectral value depended on the ratio of the length of the resonator (L) and ruby (l). The direction of displacement of the wavelength of lasing to the long wave range was determined by the thermal drift of the gain line, and the change of the mode from peak to peak was determined by the longitudinal heterogeneity of inversion.

After lasing of a single longitudinal mode in one of the peaks, the longitudinal distribution of inversion in the active rod has the following form:

where no is the initial level of inversion taking pumping into account; D is a constant coefficient. Under these conditions, the effective amplification factor α (m) as a function of the number of the mode m is proportional to the integral:

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Here B is the Einstein coefficient. Integration on the condition that the mode of the standing waves is situated at the point xo gives:

The magnitude of the spectral jump between the longitudinal modes depends both on the ratio L/l and on the position of the active rod in the resonator. In the case in which the ruby crystal is situated in the centre of the resonator and L/l = 8, the magnitude of the spectral jump between the modes was 8 inter-mode intervals [8].

It should be noted that throughout the entire lasing pulse there was unidirectional autosweeping of the wavelength of single-frequency radiation, i.e. displacement of the wavelength without the application of standing dispersion elements which induce large losses in the radiation energy. The rate of sweeping of the lasing wavelength depended on the amount by which pumping exceeded its threshold value and if this value was exceeded three times, the rate was ~0.12 nm/ms, which is four times higher than the speed of the thermal drift of the gain line. During 0.5 ms, the lasing wavelength was displaced by 60 pm and outstripped the centre of the gain line by 45 pm. The ruby laser with these spectral–time radiation characteristics is used as a highspeed spectrometer for the rapid analysis of different optical media

[8].

In the regime of non-attenuating radiation pulsations it was not

possible to obtain a stable (with respect to time) spectrum of the lasing of longitudinal modes even when using a high-quality selector represented by a Fabry–Perot resonator with the transmission factors of the mirrors of 13%. One to two longitudinal modes were excited in every lasing peak, but the lasing process was accompanied by a random change of the modes from peak to peak within the limits of the pass band of the selector, and the width of the integral radiation spectrum was approximately 5 pm (Fig 1.4e).

The appearance of non-attenuating pulsations of radiation intensity and the absence of stabilisation of the radiation spectrum are associated, as shown by the experiments, with the change of the longitudinal modes during the lasing process and not with the effect of technical perturbations

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Physics of Solid-State Lasers

Fig. 1.5 Parameters of lasing of TEMooq modes of a ruby laser with flat mirrors in normal conditions with rigid selection of the longitudinal modes in the vicinity of the lasing threshhold (a,b) and with increase of the pumping energy by 0.1% (c,d): a,c) oscillograms of radiation intensity, 20 µs markers; b,d) time evolvement of the radiation spectrum, the range of dispersion of the interferrometer 8 pm.

of the resonator. In the vicinity of the lasing threshold in the conditions with even more rigid selection of the longitudinal modes at a pass band of the selector-reference of 2 pm, where the threshold conditions were fulfilled for a single longitudinal mode, the pulsations of radiation intensity always attenuated (Fig. 1.5a, b). With increase of the pumping energy by only 0.1% (i.e. at the unchanged level of technical perturbations), after reaching the lasing of the second mode, attenuation ceased as a result of the alternation of these two modes during the lasing process (Fig 1.5c, d) [11].

The forced smoothing of the longitudinal heterogeneity of the field in the ruby crystal using compensated phase modulation (CPM) always results in stable quasistationary lasing at any excess of pumping above its threshold value (Fig. 1.6). In the first lasing peak, as under conventional conditions, a large number of longitudinal modes were excited in a wide band of the spectrum. This band was determined by the shape of the gain line and by the excess of pumping above the threshold value. However, the decrease of the width of the lasing spectrum to a single longitudinal mode in the conditions of the homogeneous field and, correspondingly, of uniform inversion in the active medium took place during a period an order of magnitude longer (Fig. 1.6c) than in the conditions of non-uniform inversion. In this case, the decrease of the width of the lasing spectrum takes place only as a result of the dispersion generated by the shape of the gain line, in accordance

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Solid-state chromium lasers in free lasing regime

Fig. 1.6 Parameters of the lasing of TEMooq modes of a ruby laser (PL2B 8×120/ 180 mm) with flat mirrors (L = 1.6 m, diameter 1.4 mm) with smoothing of the longitudinal heterogeneity of the field in ruby using condensated phase modulation in the conditions of absence of large technical interferences in the resonator (Ep = 3Et; a) oscillogram of radiation intensity; b) evolvement of the distribution of radiation intensity in the long range zone; c,d) the time evolvement of the lasing spectrum without selection of longitudinal modes (c) and with selection (d), range of dispersion of the interferrometers 24 pm (c) and 8 pm (d).

where ∆λ o is the width of the spectrum at the start of lasing, τ c is the lifetime of the photon in the resonator. After a transition process and reaching the quasistationary lasing regime, the spectral line showed inertia in relation to the thermal drift of the gain line in which stable single-frequency lasing (in the spectrum) was detected periodically over 50–100 µs. Regardless of the thermal drift of the gain line and the rearrangement of the natural modes of the resonator, the change of the longitudinal modes during the process of lasing in the conditions of the homogeneous field and inversion in the active medium took place always adiabatically and was not accompanied by the pulsations of radiation intensity.

The controlling effect of longitudinal heterogeneity of the field in the ruby crystal on the nature of lasing of TEMooq modes is clearly indicated by the following experiment. When the CPM was started with a certain delay after the start of the pumping pulse, there was always a transition in the lasing process from the regime of nonattenuating pulsations of radiation intensity to quasistationary regime (Fig. 1.7a). If the smoothing process of the heterogeneity of the field was interrupted after establishing single-frequency quasistationary lasing, this was followed immediately by a reversed transition to the regime of non-attenuating pulsations of radiation frequency (Fig. 1.7b) [8].

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