Effect of pump-induced loss on the spectral dependence of Rhodamine 6G laser efficiency under microsecond coherent pumping

As shown earlier [14, 15], at pump wavelengths within the main absorption band of various classes of dyes, the efficiency and spectral characteristics of lasing vary anomalously with pump wavelength: with increasing pump wavelength, the efficiency first rises, then drops near the peak absorption wavelength of the dye, and rises again on the long-wavelength side of its absorption band. The corresponding dependence for Rhodamine 6G is represented by curve (1) in Fig. 2. Also shown in Fig. 2 are the optical density spectrum of the Rhodamine 6G solution in ethanol [curve (2)], an analogous spectrum measured under high-power pumping [curve (4)], and the spectrum of the Einstein coefficient B_{S1 → Sn} for the absorption by excited-state singlet levels, obtained from previously reported experimental data [18, 19].

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With increasing pump wavelength (Fig. 3), the gain band extends to shorter wavelengths by 10 – 12 nm and dual-wavelength lasing is observed around the dip. The shift of the lasing spectrum to shorter wavelengths attests to an increase in wavelength-dependent optical loss [17]. Besides, such effects were shown to occur in solid-state systems as well, in particular in a two-stage dye-solution converter laser in solid matrices based on a nanoporous glass/polymer composite [20].

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To assess the possible effect of the optical loss due to the absorption by excited-state triplet levels on laser efficiency, we examined how the time variation of pump and laser pulses depended on the spectral composition of pump light (Fig. 4). In none of the cases under consideration was additional time-dependent loss detected during lasing, i.e. the loss due to the absorption of laser light by excited-state triplet levels of molecules played a secondary role in comparison with the loss due to the absorption by excited-state singlet levels. It is known from the literature that the highest laser efficiency in a Rhodamine 6G solution in ethanol under pumping by nanosecond pulses at λ_p = 530 nm can be reached at an incident power density in the range 20 – 40 MW cm⁻² and that, under the optimal conditions, the efficiency is determined by the pump and laser output losses due to the absorption by excited-state singlet levels [21].

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Comparison of the experimental data obtained using microsecond pump pulses with results for nanosecond pumping shows that the efficiency obtained at optimal energy densities in the case of ∼1-μs pump pulses (∼40 %) differs little from that in the case of nanosecond pumping at the corresponding energy densities.

To find out how the optical loss due to the absorption by excited-state singlet levels influenced laser efficiency, the Rhodamine 6G solution in ethanol was probed in the gain region of the dye (λ = 570 nm) after pumping the solution in spectral ranges around 505 (the first peak in laser efficiency), 525 (lowest laser efficiency), and 540 nm (second peak in laser efficiency). The optical density of the solution under study was measured as described above.

The following results were obtained in our experiments: Without pumping, the probe light at λ = 570 nm was absorbed (D ≈ 0.4). Under pumping near the first efficiency peak (λ = 505 nm), the probe signal at the outlet of the cuvette was amplified by 15 % to 19 %, which was due to the population inversion of the laser levels under pumping.

After pumping around the minimum in efficiency (λ_p = 525 nm), no probe signal amplification was detected. In contrast, 10 % to 12 % of the probe signal was absorbed.

After pumping around the second efficiency peak (λ_p = 540 nm), the probe signal was slightly amplified (by ∼2 %). The low value of the gain in this case was most likely the result of the loss due to the overlap of the long-wavelength part of the absorption profile with the short-wavelength part of the fluorescence spectrum.

Thus, the present experimental data lend support to the conclusion drawn previously [14, 15] that, in the case of microsecond coherent pumping of dye solutions, pump absorption leads to the formation of long-lived photoproducts (τ > 25 ns) strongly absorbing in the spectral region of amplification [21 – 23]. To account for the effect of the loss on Rhodamine 6G laser efficiency, attention should be paid to the shape of the spectrum of the Einstein coefficient B_{S1 → Sn} for the absorption by excited-state singlet levels (Fig. 2): the maximum of the spectrum coincides with the minimum in Rhodamine 6G laser efficiency.

As mentioned above, a number of effects in the gain medium of DLs can be taken into account for interpreting the present experimental data: T – T absorption, loss due to the absorption by singlet levels, pump-induced thermal inhomogeneity, and interaction of the laser output with the pump-induced dye photodestruction products [24, 25].

The loss coefficient of laser light, important for evaluating the efficiency of a coherently pumped dye laser, is directly related to beam divergence. The laser beam divergence increases as a result of pump-induced thermal defocusing. (A typical pump energy loss during dye laser operation is ∼50 %, and all this energy eventually goes to heating the gain medium, changing its refractive index.)

According to a rate equation analysis of DL kinetics under microsecond pumping [26], T – T absorption has an extremely weak impact on the effect under investigation. At the same time, electron excitation energy relaxation in higher excited singlet states was assumed to play a rather important role. A considerable contribution to this effect can be made by an induced S – S absorption, whose spectral maximum corresponds to the conventional absorption maximum [18].

Figure 5 shows an energy level diagram that allows one to describe processes in the DL with allowance for the induced absorption by S – S and T – T energy levels. In evaluating laser efficiency as a function of microsecond monochromatic pump wavelength, we took into account the spectral dependences of the absorption cross section of the S₀ – S₁ levels, σ_0S(λ), and the absorption cross section of the S₁ – S₂ levels, σ_SS(λ). The effect of the spectral dependence of the T – T absorption and related loss is taken into account by the absorption cross section σ_TT(λ).

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The system of equations describing DL operation with allowance for the above processes has the following form:

where

is the photon lifetime in the cavity; l is the cavity base length; R₁ and R₂ are the reflectivities of the mirrors; the coefficient Ω = πb²/(16Kl²) characterises the relative part of spontaneous emission propagating in the lasing channel; b is the cross-sectional size of the pump beam; the coefficient K takes into account the fraction of spontaneous emission in the laser output; N is the concentration of dye molecules; n_i is the concentration of particles in the corresponding excited states; q is the volumetric density of emitted photons at wavelength λ_g; σ_e (λ_g) is the stimulated emission cross section; σ_a (λ_g) is the absorption cross section of a higher excited singlet state; the coefficients d_ij are responsible for nonradiative relaxation of state i through an i – j transition; the coefficients P_ij are responsible for intercombination conversion of state i through this transition; τ₃₁ is the radiative lifetime of the upper laser level; n_d is the refractive index of the dye solution; I_p is pump intensity; and T_Σ is the total round-trip cavity loss.

The laser spot diameter d on the output mirror is proportional to the beam divergence θ, and the spot area S is proportional to the square of the divergence. Since the optical loss coefficient T_loss is proportional to S, we have

Taking the minimum beam divergence to be the diffraction-limited one, we obtain for the minimum output laser spot

We define the single-pass beam divergence at the output of the active laser medium as the divergence of a Gaussian beam subjected to thermal defocusing according to the equation [27]

where

is the length related to nonlinear laser beam refraction; f is the normalised width of the Gaussian beam; β is the gain coefficient; R_dif = ka²/2 is the characteristic length related to beam diffraction; k is the wavenumber; a is the laser beam radius; k is the thermal conductivity of the gain medium; ε is its dielectric permittivity; T is the absolute temperature; and x is a coordinate along the laser beam axis.

A numerical analysis indicates that, in the case of quasi-steady-state lasing, the gain coefficient β is constant and, hence, β in Eqn (4) does not vary.

Calculations using Eqn (4) show that, at constant gain medium parameters, the beam divergence increases linearly with pump power:

Here ΔP = P – P_th, where P is the pump power and P_th is the threshold pump power. In (5), the minimum divergence for our case is 0.94 mrad and the coefficient α = Δθ/ΔP is 0.82 mrad W⁻¹. From the above results, we can find the loss coefficient using relation (2):

where γ = P/P_th.

To identify the main mechanism determining the effect of pump wavelength on laser efficiency, a detailed theoretical analysis with the use of computer simulation based on an elaborate model is necessary.

In modelling a DL excited by microsecond pulses, we used parameters of the Rhodamine 6G laser dye, which is often employed in experimental studies: σ_e = 2.16 × 10⁻¹⁶ cm², τ₃₁ = 5.9 ns, λ_g = 5.6 × 10⁻⁵ cm, n_d = 1.36, N = 1.2 × 10¹⁷ cm⁻³, σ_0S = 4 × 10⁻¹⁶ cm² (at the maximum of the S₀ → S₁ absorption band), σ_TT = 5 ×10⁻¹⁷ cm² (at the maximum of the T – T absorption band), σ_SS = 1.75 × 10⁻¹⁶ cm² (at the maximum of the S₁ → S₂ absorption band), l = 20 cm, P₃₂ = 4.2 ×10⁵ s⁻¹, d₁₀ = 5 ×10¹⁰ s⁻¹, d₃₁ = 10⁷ s⁻¹, d₅₃ = 2 ×10¹⁰ s⁻¹, d₄₂ = 5 ×10⁶ s⁻¹, P₂₀ = 2 ×10⁵ s⁻¹, b = 0.5 cm, K = 10, R₁ = 1, and R₂ = 0.6.

The spectral dependences of absorption, induced absorption, and stimulated emission cross sections used in our calculations were obtained by interpolating available data [24, 25, 28 – 31].

A theoretical analysis of the effect of pumping level γ on lasing kinetics shows that the peak positions of pump and laser pulses coincide and that the laser pulse power and duration increase with γ (Fig. 6). It is seen from Fig. 6 that, with increasing pumping level, the laser pulse duration increases symmetrically with respect to the peak position, in good agreement with experimental data. At γ = 8, a laser pulse lags the leading edge of the corresponding pump pulse by ∼0.5 μs and the time it takes to reach quasi-steady-state lasing is about 0.1 μs (Fig. 6d).

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In examining DL efficiency as a function of microsecond pump wavelength, it should be taken into account that a change in pump absorption in response to pump wavelength tuning will lead to a corresponding change in threshold pump power density. Since this parameter varies, it is reasonable to assess the effect of γ on laser efficiency at a constant pump wavelength. Such data are presented in Fig. 7a. The fact that the efficiency has a maximum at γ = 5 and then gradually decreases suggests that there is a loss which rises considerably with pump power density. This is the part of pump energy which is converted into heat during laser operation and leads to thermo-optic distortion of the cavity in the form of a negative lens, whose optical power increases with pump power density [32].

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The thermal loss of energy in the gain medium is proportional to the difference between the pump and laser output energies and its effect becomes significant when the pump energy exceeds some level, E_min, determined for particular laser operation conditions. In such a case, the pump-induced negative thermal lens considerably increases the laser beam divergence. Further increase in the defocusing effect of the thermal lens eventually leads to a reduction in efficiency with increasing pump energy (Fig. 7a).

Figure 7b illustrates the experimentally observed effect of microsecond pump energy density on laser efficiency. The highest efficiency was reached at an energy density of 2 J cm⁻², whereas the threshold pump energy density was 0.5 J cm⁻², which corresponded to γ = 4, in good agreement with the calculated γ = 5.

Figure 8 shows the calculated laser efficiency as a function of pump wavelength at various normalised pumping levels. The corresponding experimental data are represented by curve (1) in Fig. 2.

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For γ ⩽ 5 (Fig. 8a), the laser efficiency as a function of pump wavelength has one maximum. To higher pumping levels there corresponds higher efficiency. In this case, the loss coefficient is relatively small because of the effect of thermal lensing. For γ > 8, the way in which laser efficiency varies with pump wavelength changes. As seen from Fig. 8b, increasing the pump power leads to a decrease in efficiency [curves (2) – (6)]. In this case, a dip is formed near the peak absorption wavelength and increases with pump power [curves (3) – (6)]. The thermal lensing-induced loss coefficient is here rather large. It reaches the highest value at the peak absorption wavelength of the dye because at this wavelength the largest fraction of the absorbed energy goes to thermo-optic distortion of the gain medium. The calculation results are well supported by the experimental data presented in Fig. 2 [curve (1)], which clearly shows a dip in Rhodamine 6G laser efficiency under pumping at λ_p = 530 nm.

Experimental evidence for thermo-optic distortion and the formation of a negative lens in the Rhodamine 6G solution in ethanol under microsecond pumping is presented in Fig. 9a, which shows a photograph of the far-field spatial intensity distribution under pumping at λ_p = 525 nm, near the minimum in laser output energy (γ = 8). It is clearly seen that there is an interference pattern in the form of concentric circles, due to the thermal lensing in the solution. To the observed picture there corresponds an angular divergence θ ≈ 10⁻³ rad. For comparison, Fig. 9b shows a spatial laser output intensity distribution under nanosecond pumping (λ_p = 532 nm). Comparison demonstrates significant differences in the structure and size of spots and, accordingly, a smaller angular divergence in the case of nanosecond pumping.

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It is worth noting that pump-induced thermal lensing is not the only cause of the decrease in laser efficiency as a function of pump wavelength of various dyes, because experimentally observed dips were located not at the centre of their absorption bands. Dips in the spectral dependence of laser efficiency were located both in the centre of the absorption bands and on their short- or long-wavelength wings [14, 15, 23]. To understand this, it should be taken into account that the spectral absorption and fluorescence profiles of the dye are inhomogeneously broadened. Under high-power monochromatic pumping, the absorption and gain profiles undergo a certain transformation due to the corresponding changes in the population of energy states. We are thus led to conclude that the position of the dips is mainly determined by the overall effect of two effects considered in this study: nonlinear refraction and relaxation processes in the system of singlet levels.

A numerical analysis of the influence of induced absorption and the corresponding loss in the channel of higher singlet levels shows that this loss is relatively small because of the small population of the corresponding energy states. The relative population of the S₂ state is ∼10⁻⁶, and its lifetime is a few picoseconds.