Semiclassical theory of transient intracavity stimulated

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Semiclassical theory of transient intracavity stimulated Raman scattering in compact lasers

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Journal of Physics B: Atomic, Molecular and Optical Physics J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 105402 (14pp)

doi:10.1088/0953-4075/47/10/105402

Semiclassical theory of transient intracavity stimulated Raman scattering in compact lasers S V Voitikov 1 , A A Demidovich 2 , M B Danailov 2 and V A Orlovich 1 1 2

Institute of Physics, National Academy of Sciences of Belarus, Minsk 220062, Belarus LaserLab ELETTRA-Sincrotrone, Trieste I-34012, Italy

E-mail: [email protected] Received 22 January 2014, revised 7 April 2014 Accepted for publication 8 April 2014 Published 7 May 2014 Abstract

We have developed a semiclassical theory of transient intracavity stimulated Raman scattering (SRS) of the fundamental (laser) wave into Stokes and anti-Stokes waves of several orders in compact lasers. From the analysis of the nonlinear wave equation for the field in the laser cavity and the constitutive equations for the Raman medium a hierarchy of rate equations for the amplitudes of the fundamental and Stokes/anti-Stokes waves, amplitudes of collective vibrations and populations of excited states of the Raman medium has been obtained. It has been shown that the known semiclassical and phenomenological models of intracavity SRS-conversion are special cases of the proposed theory. We have used the theory to calculate the output pulse parameters of a solid-state Raman microchip laser in which we obtained experimentally, along with high peak power first Stokes pulses, also first anti-Stokes and second Stokes radiation but only in the spectrum. The results of modelling agree well with the experimental data and show that anti-Stokes radiation is due to the multiwave mixing. Keywords: Raman lasers, stimulated Raman scattering, solid-state lasers, intracavity multiwave mixing, laser theory (Some figures may appear in colour only in the online journal)

cross-sections, and the sufficiently high efficiency of SRSconversion, the conditions for Raman medium excitation and anti-Stokes wave generation due to both the parametric and two-photon processes could, in principle, also be realized. The intracavity multiwave mixing will make an additional contribution to the rate of rise of induced polarization of the Raman medium radiating at the frequency of the second and higher-order Stokes waves as well as at the frequency of antiStokes waves and, therefore, could decrease their thresholds. Thus, for intracavity SRS-conversion of the fundamental wave into one or several frequency-shifted waves compact lasers are, in a specified sense, unique solid-state based laser systems, in which the effects of high-intensity transient/quasi-stationary SRS-conversion can manifest themselves simultaneously and versatilely. These are the generation of Stokes/antiStokes waves of several orders, the time-dependent (not instantaneous) processes of induced polarization relaxation

1. Introduction Due to the rapid development of compact solid-state Raman lasers the effect of stimulated Raman scattering (SRS) received new opportunities to demonstrate its features and advantages. Recent investigations have shown that intracavity SRS frequency conversion makes it possible to obtain Stokes pulses in compact lasers with durations from 50–180 ps to 18–20 ns and energies from unities to several dozens of microjoules [1–14]. Attaining a high intracavity peak power in the first Stokes wave in the cavity makes the generation possible in such lasers of pulses of second [15, 16], third [17], and, possibly, higher-order Stokes waves, to whose rise and amplification not only the two-photon processes but also the parametric multiwave mixing of the fundamental (laser) and Stokes/anti-Stokes waves will contribute. In compact lasers, because of the high intracavity powers, the small beam 0953-4075/14/105402+14$33.00

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J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 105402

for the amplitudes of collective vibrations defining the nonlinear polarization induced in the Raman medium as a source of stimulated radiation with shifted frequencies and equations for the populations of Raman-active centres in the excited state. Section 3 considers the corollaries of the developed semiclassical theory and special cases. It has been shown that the existing semiclassical and phenomenological models of intracavity SRS-conversion are special cases of the semiclassical theory developed here. A brief summary and conclusions are presented in section 4. In the appendix the mode overlap integrals are presented.

(dephasing) of the Raman medium, Raman medium excitation, and transient intracavity multiwave mixing. All this calls for the development of a theory of intracavity SRS adequately taking into account the above features as well as the features imposed on the generation by the compact design of such lasers. In the majority of the existing phenomenological models, the intracavity SRS was described in terms of the field intensity or the number (density) of photons and only the quasi-stationary approximation for the Raman process was considered, which is valid in the case where generated pulses are much longer than the dephasing time of the Raman medium [1, 2, 4–8, 12, 13]. Moreover, the above phenomenological models do not allow taking into account the Raman medium excitation and multiwave mixing. The known semiclassical models of Raman lasers proposed earlier in [18–21] describe the intracavity SRS in terms of the wave and collective vibration amplitudes and the approaches used in them can deal only with quasi-stationary SRS and quasistationary multiwave mixing and do not allow the dephasing and excitation of the Raman medium to be correctly taken into account. Recently in [22, 23] a quantum theory of intracavity SRS-conversion in lasers was proposed. As is known (see, e.g., [24]), the quantum description of optical processes is based on the formulation of the field + matter Hamiltonian and subsequent derivation of dynamic equations. In [23] we applied this approach to intracavity SRS-conversion and obtained quantum Heisenberg–Langevin equations for the amplitudes of laser and Stokes/anti-Stokes waves and equations for matter describing the dynamics of induced nonlinear polarization and excitation of the Raman medium. As is known, in nonlinear optics, along with a quantum description of the field and matter, also a semiclassical description is widely used for the cases where the specific quantum effects are of no importance [24]. In contrast to the most comprehensive quantum approach based on the analysis and transformation of the Hamiltonian, the semiclassical approach is based on the analysis of a nonlinear wave equation, it operates with classical terms and, therefore, can be much more visual and demonstrative. In [15–17], for numerical modelling of experimental studies of the twoand three-Stokes generation in compact lasers, a model of transient intracavity SRS-conversion without Raman medium excitation was developed. The model is based on going from quantum equations for the operators of physical quantities [23] to equations for their quantum–mechanical averages. A satisfactory agreement with the experimental results was obtained. In the present paper, we propose a theory of transient intracavity SRS of the fundamental wave into Stokes and antiStokes waves, including those of several orders, accompanied by transient multiwave mixing and Raman-medium excitation. In section 2, from the analysis of the nonlinear wave equation for the field and the Raman medium in the cavity, we derive equations for the slowly varying amplitudes of the fundamental (laser) wave and Stokes/anti-Stokes waves of several orders. Then we obtain a hierarchy of equations

2. Theory In this section, the semiclassical theory of transient intracavity SRS is developed for lasers whose cavity contains an active medium, irradiating via stimulated transitions at the fundamental wavelength, a Raman medium converting fundamental radiation into Stokes and/or anti-Stokes radiation, and a saturable absorber (if pulsed performance of the laser is required). The cavity can be common for all generated and SRS-converted waves, or it can be coupled with an additional cavity for the Raman medium within a common cavity (coupled-cavity laser configuration). The model assumes that the optical waves in a cavity can be approximated by standing waves. As is known [25], for solid-state lasers this approximation works not only in the case of one-pass low gain/absorption, but also gives good results for the lasers with solid-state absorbers where initial transmittance is not less than ∼60%. This is due to the fact that during the initiation and generation of a laser pulse the most commonly used crystalline absorbers become saturated, so their transmittances reach the values from 90% to 99% [26] and optical losses in absorbers become very small. We expand the field in the cavity into a series of spatial cavity modes formed from counterrunning TEM00 Gaussian beams [23, 27, 28]: ⎫ ⎧ ⎬ 2 ⎨ 1 uλ(m) (r, z), z1(m) < z < z2(m) , Uλ (r, z) = ⎭ Lλ ⎩ μλ(m)

2 1 r2 exp − uλ(m) (r, z) =

2 (m) π rλ (z) rλ(m) (z) r2 (m) (m) (m) (m) (m) − ψλ (z) + ϕλ × sin kλ z − z0,λ + kλ , 2Rλ(m) (z) (1) where λ is the mode number (number of half-wavelengths present on the cavity length, λ = 1, 2, 3, . . . , ∞) for the cavity at a wavelength λλ , r is the transverse radius, z is the longitudinal coordinate, the superscript m denotes the media or the air gaps (am stands for the active medium, sa—for the saturable absorber, Rm—for the Raman medium, v—for the gaps between the laser intracavity elements), and 2

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kλ(m) = μλ(m) 2π /λλ is the wave number of the λ-mode in the m-medium, ⎡ ⎤1/2 (m) 2 z − z 0,λ ⎦ , rλ(m) (z) = rλw ⎣1 + (2) (m) zR,λ

where Vcav is the cavity volume. The wave equation for the field in the cavity is of the form 2 r, t ) − 1 ∂ ∇ 2 E( c2 ∂t 2

(Rm) (r, t ) + P (am) (r, t ) + P (sa) (r, t ) × E(r, t ) + 4π P lin lin lin 4π ∂ 2 (Rm) (am) (r, t ) + P (sa) (r, t )], = 2 2 [P (r, t ) + P (8) c ∂t (Rm) (r, t ) is where c is the speed of light in free space, P the nonlinear polarization induced in the Raman medium, (am) (r, t ) is the nonlinear polarization of the active P medium causing stimulated emission at the fundamental (sa) (r, t ) is the nonlinear polarization of the wavelength, P (Rm) (r, t ), saturable absorber describing the light absorption, P lin (am) (sa) (r, t ) are the linear polarizations of the P r, t ), and P lin ( lin corresponding media. For the linear polarizations we have (m) 1 (m) (r, t ) == P αλ ∈λ (Eλ (t ) e−iωλ t + Eλ∗ (t ) eiωλ t )Uλ (r, z), lin 2 λ

(m) is the longitudinal rλw is the waist radius of the λ-mode, z0,λ coordinate of the waist of the λ-mode in the m-medium (or the longitudinal coordinate of the waist image if the waist is (m) located outside the m-medium), and zR,λ is the Rayleigh range for the λ-mode in the m-medium, 2 μλ(m) (m) = π rλw (3) zR,λ λλ (m) 2 zR,λ (m) (m) Rλ (z) = z − z0,λ + , (4) (m) z − z0,λ (m) z − z0,λ (m) ψλ (z) = arctan , (5) (m) zR,λ

of the m-medium at the where μλ(m) is the refractive index (m) (m) is the optical length wavelength λλ , Lλ = m μλ L (m) of the cavity at the wavelength λ , L is the m-medium λ √ length, 2/Lλ is the normalization factor, ϕλ(m) is the phase incursion determining the matching conditions of the field distributions at the boundaries of media with different refractive indices, z1(m) and z2(m) are the longitudinal coordinates of the m-medium faces, respectively. In expression (1), the (m) factor 1/ μλ provides spatial continuity of the intensity between neighbouring AR-coated elements of the laser. For the configuration of lasers with a common cavity, the spatial mode distribution (expression (1)) for each generated wave has terms corresponding to all intracavity elements (media) and gaps. In the case of the coupled-cavity laser, expression (1) for the fundamental wave still has terms corresponding to all intracavity elements (media) and gaps, while for the Stokes/anti-Stokes waves it has only two terms corresponding to the Raman medium (m → Rm) and gaps (m → v) in the Raman cavity. Below we use terminology and notations close to those introduced in the review [29]. Once the spatial modes have been determined (expression (1)), we can expand the field in the laser cavity in the form, 1 r, t ) = ∈λ (Eλ (t ) e−iωλ t + Eλ∗ (t ) eiωλ t )Uλ (r, z), (6) E( 2 λ

(9) αλ(m)

where is the linear polarizability of the m-medium at the wavelength λ

2 to the corresponding refractive index by λ related the relation μλ(m) = 1 + 4π αλ(m) . Below we consider the field and matter as isotropic. For the second-order time derivatives in the rotating wave and slowly varying amplitude approximations in equation (8) and for the spatial Laplacian ∇ 2 also in the paraxial approximation (see, e.g., [25]) in equation (8) the following substitutions should be made (Rm)

1 ∂2 (am) (r, t ) + P (sa) (r, t ) E(r, t ) + 4π P r, t ) + P lin ( lin lin 2 2 c ∂t μλ (z)2 1 dEλ (t ) −iωλ t e −2iωλ → 2 c 2 dt λ + ωλ2 Eλ (t )e−iωλ t + c.c. Uλ (r, z), z) → − ∇ 2 E(r,

(10) where the refractive index is presented as a step function of the longitudinal coordinate μλ (z) with values μλ(m) in the m-medium and 1 in the cavity air gaps, c.c. means ‘complex conjugate’. Once substitutions (10) in equation (8) have been made, we multiply the left- and the right-hand sides of equation (8) by Uλ (r, z) and integrate it over the Ramanmedium volume, VRm, to obtain in the isotropic case the following expression for the rate of Eλ (t ) evolution for the isotropic case 1 2 dEλ (t ) = iωλ 4π uλ (r, z)Pλ (r, t ) d3r dt 2 μλ Lλ VRm dEλ (t ) dEλ (t ) dEλ (t ) + + , (11) + dt am dt sa dt loss where dEλ (t )/dt|am and dEλ (t )/dt|sa are determined by the nonlinear polarizations P(am) (r, z) and P(sa) (r, z), and represent the rates of change in time the wave amplitude due to the

where Eλ (t ), ∈λ and ωλ are the slowly varying amplitude, the unit vector of polarization, and the cyclic frequency of a wave with wavelength λλ , respectively. The normalization in equation (1) was chosen such that the expression for the field energy Wλ (t ) at frequency ωλ has no spatially dependent factors, namely μ2λ (z) Wλ (t ) = |E(r, t )|2 d3r Vcav 4π 1 ∗ = Eλ (t )Eλ (t )μ2λ (z)Uλ2 (r, z) d3r Vcav 8π =

1 ∗ E (t )Eλ (t ), 8π λ

μλ (z)2 1

ωλ2 Eλ (t ) e−iωλ t + c.c. Uλ (r, z), 2 c 2 λ

(7) 3

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amplification in the active medium and the absorption in the saturable absorber (if any), respectively, and dEλ (t )/dt|loss is both the intracavity passive and mirror loss. Below we describe only the processes in the Raman medium and, for simplicity, will temporarily omit the superscript (Rm) as we already did in equation (11). Equation (11) holds for all waves generated in the laser cavity; these are the fundamental wave and the Stokes and/or anti-Stokes waves including high-order ones. The number of generated Stokes and/or anti-Stokes waves is determined by the experimental realization of a given laser and, in principle, can vary from one to several Stokes and/or anti-Stokes waves. In the semiclassical approach [29], the induced nonlinear polarization of the Raman medium is ∂α (Rm) q(r, t )E(r, t ), (12) (r, t ) = N P ∂q where q(r, t ) is the amplitude of local vibrations of Ramanactive centres, N is the density of Raman-active centres, and (∂α/∂q) is the derivative of the Raman-medium polarizability with respect to the space coordinate. In this paper, the effects of the non-resonant third-order susceptibility are not taken into account. In most Raman-active crystals providing effective Raman scattering in the near IR region, such as barium nitrate, diamond, vanadates, tungstates, molybdates, etc, the nonlinear refraction coefficients (being proportional to non-resonant third-order susceptibilities), do not exceed the value of n2 ≈ 2 10−15 cm2 W−1 [30–32]. Accordingly, for the intracavity intensities 5 GW cm−2 that is realized very often in compact solid-state Raman lasers the corrections for refractive indices (product n2-intensity) are less than 10−5. It means the effects of non-resonant third-order susceptibilities can be neglected. Using the expansion 1 q(r, t ) = (13) (qν (r, t )e−iων t + q∗ν (r, t )eiων t ) 2 ν

On the right-hand side of equation (15) and in all equations given below, variables with subscripts λ+ν and λ−ν pertain to waves with frequencies ωλ+ν = ωλ + ων and ωλ−ν = ωλ − ων , respectively. So, relative to the wave with frequency ωλ (λwave) waves with frequencies ωλ+ν ((λ + ν)-wave) and ωλ−ν ((λ−ν)-wave) are anti-Stokes and Stokes waves, respectively. The equations for the local vibration amplitude q(r, t ) and the excited-state population n(r, t ) of Raman-active centres were obtained from quantum–mechanical description of Raman matter (see, e.g., [29] and references therein) and have the form: 2 dq(r, t ) d2 q(r, t ) + ω02 q(r, t ) + 2 dt T2 dt 1 ∂α E(r, t )2 [1 − 2n(r, t )], (16) = 2m ∂q and ∂α dn(r, t ) 1 dq(r, t ) 1 + E(r, t )2 , n(r, t ) − n¯ = dt T1 2ω0 ∂q dt (17) where ω0 is the resonant cyclic excitation frequency of Ramanactive centres, n¯ is the population of Raman-active centres in thermal equilibrium, m is the mass of Raman-active centres, T1 , and T2 are the relaxation time of excited Raman-active centres and the dephasing time of local vibrations, respectively. Using the slowly varying amplitude approach, after substituting expression (1) into equation (16), we obtain the rate equation 2ων i dqν (r, t ) ω02 − ων2 − i qν (r, t ) + dt 2ων T2 ∂α i [1 − 2n(r, t )] = 4mω ∂q ν 1 ∗ × 2Eκ (t )Eκ−ν (t ) √ μκ Lκ κ 1 ×√ uκ (r, z)uκ−ν (r, z) , (18) μκ−ν Lκ−ν where qν (r, t ) is the slowly varying amplitude of vibration at where on the right-hand side summation over the subscript the cyclic frequency ων , for the nonlinear polarization (12) we κ is performed with respect to all waves generated and/or obtain the following expression: SRS-converted in the laser. For instance, if in a given laser a fundamental, two Stokes and one anti-Stokes waves are 1 ∂α (qν (r, t ) e−iων t + q∗ν (r, t ) eiων t ) generated, then the subscript κ in the respective summation P(Rm) (r, t ) = N ∂q 2 ν takes on such values that amplitudes Eκ (t ) correspond √ 1 2 sequentially to the first anti-Stokes, the fundamental, and −iωκ t ∗ iωκ t (Eκ (t ) e + Eκ (t ) e ) √ uκ (r, z) . × 2 first Stokes waves, while the amplitudes Ek−ν (t ) sequentially μ L κ κ κ (14) correspond to the fundamental, and the first and second Stokes waves, respectively. Accordingly, there are pair amplitude If there is only one frequency of local vibrations ων in the products of the first anti-Stokes and the fundamental, the Raman medium, then after the substitution of expression (14) fundamental and the first Stokes, the first and the second into equation (11) for the rate of change in the amplitude Eλ (t ) Stokes waves in the sum between curly brackets on the rightwe obtain hand side of equation (18), and all these waves make their own 1 ∂α dEλ (t ) 1 contribution to the rise/damping of local vibrations of Raman= iωλ 4π N active centres at frequency ων . If higher-order Stokes/antidt 2 2 ∂q Stokes waves are also generated, then the pair products of the 1 1 × 2√ uλ (r, z) q∗ν (r, t )Eλ+ν (t ) √ uλ+ν (r, z) corresponding wave amplitudes are added to the sum over the μλ+ν Lλ+ν μλ Lλ VRm subscript κ in equation (18). 1 Let us introduce renormalized amplitudes of collective uλ−ν (r, z) d3r + qν (r, t )Eλ−ν (t ) √ μλ−ν Lλ−ν vibrations of the Raman medium Qλ,λ−ν (t ) defining them as dEλ (t ) dEλ (t ) dEλ (t ) + + + . (15) projections of local vibration amplitudes qν (r, z) onto the pairs dt am dt sa dt loss of spatial modes within the Raman medium: 4

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∂α 4π Qλ,λ−ν (t ) = Qλ−ν,λ (t ) = −i N ∂q × qν (r, t )uλ (r, z)uλ−ν (r, z) d3r,

dN{2p} (t ) 1 0 + N{2p} (t ) − {2p} n¯ dt T1 1 ων 1 1 = ω0 N 8π β μβ Lβ μβ−ν Lβ−ν ∗ ∗ Eβ (t )Eβ−ν (t )Q{2p};β,β−ν (t )+Eβ∗ (t )Eβ−ν (t )Q{2p};β,β−ν (t ) , × 2 (25)

(19)

VRm

where, unlike the definition of amplitudes of collective vibrations used in [29], the renormalization factor put in parentheses is added. This allows putting the equations for the amplitude of collective vibrations and the excited-state populations of Raman medium into a more graphical form. Therefore, accordingly to (19), the rate equation for Eλ (t ) has the form dEλ (t ) 1 1 1 Q∗ (t )Eλ+ν (t ) = ωλ dt 2 μλ Lλ μλ+ν Lλ+ν λ+ν,λ 1 1 − Qλ,λ−ν (t )Eλ−ν (t ) μλ Lλ μλ−ν Lλ−ν dEλ (t ) dEλ (t ) dEλ (t ) + + + . (20) dt dt dt am

sa

where the renormalized collective vibration amplitudes Q{2p};β,β−ν (t ) (standing for Qλ,λ−ν;κ,κ−ν;β,β−ν (t )) are projections of the local vibration amplitude qν (r, z) onto six corresponding spatial modes within the Raman medium: ∂α 4π Qλ,λ−ν;κ,κ−ν;β,β−ν (t ) = −i N qν (r, t ) ∂q VRm × uλ (r, z)uλ−ν (r, z)uκ (r, z)uκ−ν (r, z)uβ (r, z)uβ−ν (r, z) d3r. (26) Summation with respect to β on the right-hand side of equation (25) is performed over all waves generated in the laser and frequency-converted in the Raman medium. Thus, the excitation, as well as collective vibrations of the Raman medium resulting from the Raman scattering of the λ-wave into the (λ − ν)-wave, are due to the interaction of not only these two waves, but also of all the other waves for which, in a given laser, the conditions for their intracavity amplification are realized. The rate equation for the amplitude Q{3p} (t ) (standing for Qλ,λ−ν;κ,κ−ν;β,β−ν (t )) is derived from equation (18) in a similar manner as was done above for the amplitude Qλ,λ−ν (t ) with a smaller number of spatial modes in the corresponding integrand:

ω0 2 1 μτ μτ −ν Eτ (t )Eτ∗−ν (t ) dQ{3p} (t ) = c gτ −ν dt ων T2 τ Lτ Lτ −ν 8π ωτ −ν 0 × L(Rm) {3p};τ,τ −ν − 2N{3p};τ,τ −ν (t )

i 2 1 − ω0 − ων2 Q{3p} (t ) − Q{3p} (t ), (27) 2ων T2 where 4 0 = uλ (r, z)uλ−ν (r, z) λ,λ−ν;κ,κ−ν;β,β−ν;τ,τ −ν L(Rm) VRm × uκ (r, z)uκ−ν (r, z)uβ (r, z)uβ−ν (r, z)uτ (r, z)uτ −ν (r, z) d3r, (28) 4 n(r, t ) Nλ,λ−ν;κ,κ−ν;β,β−ν;τ,τ −ν (t ) = (Rm) L VRm × uλ (r, z)uλ−ν (r, z)uκ (r, z)uκ−ν (r, z)uβ (r, z)uβ−ν (r, z) (29) × uτ (r, z)uτ −ν (r, z) d3r.

loss

Multiplying the left- and right-hand sides of equation (18) by ((−4π i/)N(∂α/∂q))uλ (r, z)uλ−ν (r, z) and integrating them over the Raman medium volume, we obtain the rate equation for the renormalized amplitudes of collective vibrations

1 i 2 dQλ,λ−ν (t ) ω0 − ων2 Qλ,λ−ν (t ) + Qλ,λ−ν (t ) + dt 2ων T2

∗ (t ) μκ μκ−ν Eκ (t )Eκ−ν ω0 2 1 = c gκ−ν ων T2 κ Lκ Lκ−ν 8π ωκ−ν 0 (Rm) λ,λ−ν;κ,κ−ν − 2Nλ,λ−ν;κ,κ−ν (t ) , (21) ×L where the gain gκ−ν corresponding to the SRS-conversion of the κ-wave into the (κ − ν)-wave is [29] 2 4π 2 T2 ωκ−ν ∂α gκ−ν = N , (22) 2 μκ μκ−ν c mω0 ∂q 4 0 λ,λ−ν;κ,κ−ν = (Rm) uλ (r, z)uλ−ν (r, z)uκ (r, z) L VRm (23) × uκ−ν (r, z) d3r, Nλ,λ−ν;κ,κ−ν (t ) =

4

n(r, t )uλ (r, z)uλ−ν (r, z) L(Rm) VRm (24) × uκ (r, z)uκ−ν (r, z) d3r. On the right-hand side of equation (21) in the summation over κ, the terms with κ = λ describe the cascade (two-photon) mechanism of SRS-conversion of the λ-wave only into the (λ − ν)-wave, and the terms with κ = λ describe the wave generation by multiwave mixing of fundamental and Stokes and/or anti-Stokes waves of one or several orders. To avoid very long subscripts in equations we introduce the notations {2p},{3p},{4p} etc for the sets composed of two, three, four etc pairs of subscripts. For instance, N{2p} (t ) stands for Nλ,λ−ν,κ,κ−ν (t ), Q{2p};β,β−ν (t ) is related to Q{3p} (t ) and stands for Qλ,λ−ν;κ,κ−ν;β,β−ν (t ) etc. For populations N{2p} (t ), which are the projections of the local population n(r, t ) of excited Raman-active centres onto four corresponding spatial modes within the Raman medium, in the slowly varying amplitude approximation we get from equation (17) the following rate equation:

The rate equation for N{4p} (t ) (standing for Nλ,λ−ν;κ,κ−ν;β,β−ν;τ,τ −ν (t )) is derived from equation (17) in a similar manner as was done for N{2p} (t ), dN{2p} (t ) 1 0 + N{2p} (t ) − {2p} n¯ dt T1 ων 1 1 1 = ω0 N 8π ρ μρ Lρ μρ−ν Lρ−ν ∗ ∗ Eρ (t )Eρ−ν (t )Q{2p};ρ,ρ−ν (t )+Eρ∗ (t )Eρ−ν (t )Q{2p};ρ,ρ−ν (t ) , × 2 (30) 5

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on the analysis and transformation of the field + matter Hamiltonian [23]. This is achieved when in the quantum equations the frequency Stark shifts, the self-energy terms, and the quantum noise are neglected, and the operator equations are reformulated into equations for quantum–mechanical averages. Below we illustrate the proposed theory with the example of the simplified and, therefore, more graphical case of transient intracavity SRS conversion of the fundamental wave into two Stokes and one anti-Stokes waves with Ramanmedium excitation limited to the first order, i.e., we neglect all populations of higher orders such as N{4p} (t ). For clearness, instead of subscripts λ, λ − ν, κ, κ − ν, β, β − ν, τ , and τ − ν we will use subscripts A1, L, S1 and S2, corresponding to the first anti-Stokes, the fundamental, the first and second Stokes waves, respectively. From equation (20) for the wave amplitudes we obtain the following equations: 1 dEA1 (t ) 1 QA1 ,L (t )EL (t ) = − ωA1 ! √ dt 2 μA1 LA1 μL LL dEA1 (t ) dEA1 (t ) + + , (34) dt dt

where Q{2p},ρ,ρ−ν (t ) = Qλ,λ−ν;κ,κ−ν;β,β−ν;τ,τ −ν;ρ,ρ−ν (t ) 4π ∂α = −i N qν (r, t )uλ (r, z)uλ−ν (r, z) ∂q VRm × uκ (r, z)uκ−ν (r, z)uβ (r, z)uβ−ν (r, z)uτ (r, z)uτ −ν (r, z) (31) × uρ (r, z)uρ−ν (r, z) d3r are projections of the local vibration amplitude qν (r, t ) onto already ten spatial modes. Deriving in such a way the rate equations for renormalized amplitudes of collective vibration and populations of excited Raman-active centres as projections of the corresponding local vibration and population onto two, three, four, or more pairs of spatial modes in the Raman medium, we obtain a hierarchy of laser rate equations describing the transient intracavity SRS of the fundamental wave into Stokes and/or anti-Stokes waves of one or several orders. Such a hierarchy of equations corresponds to the expansion used in perturbation theories and terminates when upon the addition of new rate equations the result changes negligibly. If the amplitude of collective vibration of the Raman medium following amplitude (31) in the hierarchy of equations is so small that the value of its corresponding projection of n(r, t ) onto products of twelve modes is negligibly small, then the equation for amplitude (31) is the last one in the hierarchy of equations and has the following form:

ω0 c2 μχ μχ−ν dQ{5p} (t ) = gχ−ν dt ων T2 χ Lχ Lχ−ν ×

∗ Eχ (t )Eχ−ν (t )

8π ωχ−ν

0 L(Rm) {5p},χ,χ−ν −

sa

loss

1 dEL (t ) 1 Q∗A ,L (t )EA1 (t ) = ωL ! √ dt 2 μA1 LA1 μL LL 1 1 ! −√ QL,S1 (t )ES1 (t ) μL LL μS1 LS1 dEL (t ) dEL (t ) dEL (t ) + + + , dt dt dt

1 Q{5p} (t ), T2 (32)

am

dES1 (t ) 1 = ωS1 dt 2

where 0 0 {5p},χ,χ−ν = λ,λ−ν;κ,κ−ν;β,β−ν;τ,τ −ν;ρ,ρ−ν;χ,χ−ν 4 = (Rm) uλ (r, z)uλ−ν (r, z) L VRm × uκ (r, z)uκ−ν (r, z)uβ (r, z)uβ−ν (r, z)uτ (r, z) (33) × uτ −ν (r, z)uρ (r, z)uρ−ν (r, z) d3r.

−!

√

sa

1 ! Q∗L,S1 (t )EL (t ) μL LL μS1 LS1

1 !

QS1 ,S2 (t )ES2 (t ) μS1 LS1 μS2 LS2 dES1 (t ) dES1 (t ) + + , dt sa dt loss dES2 (t ) 1 1 ! = ωS2 ! Q∗S ,S (t )ES1 (t ) dt 2 μS1 LS1 μS2 LS2 1 2 dES2 (t ) dES2 (t ) + + . dt dt

The formulation of equations (20), (21), (25), (27), (30), (32) is the principal result of the present work devoted to the development of a semiclassical theory of transient SRSconversion in compact lasers. These equations describe the transient (and, as a special case, quasi-stationary) intracavity SRS-conversion of the fundamental wave into Stokes and anti-Stokes waves of one or several orders, the dynamics of initiation, amplification and dephasing of induced polarization (collective vibrations) of the Raman medium and its excitation as a result of the interaction of the Raman medium with all waves generated in the laser.

(35)

loss

sa

(36)

(37)

loss

From equation (21) for the amplitudes of collective vibrations Qλ,λ−ν (t ) (the subscripts (λ, λ − ν) stand for (A1, L), (L, S1) or (S1, S2)) we have the equations

ω0 2 1 1 (Rm) gL μA1 μL dQλ,λ−ν (t ) = L c EA (t )EL∗ (t ) dt ων T2 8π ωL LA1 LL 1 0 × λ,λ−ν;A − 2Nλ,λ−ν;A1 ,L (t ) 1 ,L

0 gS1 μL μS1 EL (t )ES∗1 (t ) λ,λ−ν;L,S − 2Nλ,λ−ν;L,S1 (t ) + 1 ωS1 LL LS1

0 gS μS1 μS2 + 2 ES1 (t )ES∗2 (t ) λ,λ−ν;S − 2N (t ) λ,λ−ν;S1 ,S2 1 ,S2 ωS2 LS1 LS2

i 2 1 − ω0 − ων2 Qλ,λ−ν (t ) − Qλ,λ−ν (t ), (38) 2ων T2

3. Discussion It can be shown that the semiclassical equations (20), (21), (25), (27), (30), (32) obtained from the wave equation for the field in the laser cavity are fully consistent with the equations of quantum theory of intracavity Raman conversion based 6

S V Voitikov et al


coupling factors δ... depend only on the dispersion of refractive indices and are independent of the beam radii. The values of coupling factors of multiwave mixing define the intensity of multiwave mixing in the Raman medium. The corresponding expressions for σ...0 and δ... are presented in the appendix (see equations (A.9)–(A.28)). Note that for the two-photon (cascade) processes, the coupling factors of multiwave mixing are identically equal to 1. Equations (38)–(40) demonstrate the features of the multiwave mixing under intracavity SRS-conversion and the energy redistribution between the generated/converted waves when the excitation of the Raman medium is not negligible. As is seen from equations (38)–(40), induced polarization of the Raman-medium emitting at the frequency of the first Stokes wave is created not only under Raman scattering of the fundamental wave into a first Stokes one, but also under Raman scattering of the fundamental wave into a first anti-Stokes wave and under Raman scattering of the first Stokes wave into a second Stokes wave, etc. Similarly induced polarization of the Raman medium emitting at the frequency of the second Stokes wave is created not only under Raman scattering of the first Stokes wave into a second Stokes wave, but also under Raman scattering of the fundamental wave into a first Stokes and first anti-Stokes waves, etc. If for a particular laser with more or less sufficient excitation of the Raman medium the cavity and laser element parameters are chosen such that, for example, δL,S1 ;S1 ,S2 = 0, then even in this case the effect of multiwave mixing will take place. According to equations (39), (40), it is realized directly through the renormalized population NL,S1 ;S1 ,S2 (t ) and indirectly through the renormalized populations NL,S1 ;L,S1 (t ), NA1 ,L;A1 ,L (t ) and NS1 ,S2 ;S1 ,S2 (t ). So, for intracavity Raman conversion a clear line between the twophoton Raman conversion and the conversion resulting from the multiwave mixing can only be drawn if the Raman-medium excitation is negligibly small. If the Raman medium excitation is more or less sufficient, then the two-photon intracavity SRS will depend explicitly on the multiwave mixing, and vice versa, the generation of Stokes/anti-Stokes waves by multiwave mixing will depend explicitly on the two-photon processes. Let us consider some special cases of the proposed theory. Neglecting the excitation of the Raman medium leads to the models of intracavity transient SRS-conversion proposed in [15, 16] and [17] for analyzing and calculating the output parameters of two- and three-Stokes generation in solid-state Raman lasers, respectively. In these works, the modelling results agreed well with various and detailed experimental data. If SRS-conversion occurs in the quasi-stationary mode (i.e., the characteristic times of change in the amplitudes of generated/converted waves is much longer than the dephasing time (T2) of the Raman medium), we obtain other special cases of the proposed theory. In the quasi-stationary mode we obtain, in accordance with equations (34)–(40), the following equations for wave amplitudes: ωA ω0 2 (Rm) 1 1 dEA1 (t ) =− 1 cL dt 16π ων 1 − iωT2 μA1 μL gL μA1 μL × EA (t )EL∗ (t )EL (t ) ωL LA1 LL 1

where the noise terms are omitted to make formulas shorter. According to the definitions (23), (24), on the right-hand side of equation (38) the order of subscripts in overlap integrals and populations is arbitrary. For instance, when λ stands for 0 S1 and λ − ν stands for S2 then λ,λ−ν;A = S01 ,S2 ;A1 ,L = 1 ,L 0 0 0 0 A1 ,L;S1 ,S2 , λ,λ−ν;L,S1 = S1 ,S2 ;L,S1 = L,S1 ;S1 ,S2 , Nλ,λ−ν;A1 ,L (t ) = NS1 ,S2 ;A1 ,L (t ) = NA1 ,L;S1 ,S2 (t ), Nλ,λ−ν;L,S1 (t ) = NS1 ,S2 ;L,S1 (t ) = NL,S1 ;S1 ,S2 (t ), etc. For the population N{2p} (t ) (standing for Nλ,λ−ν;k,k−ν (t )), where the subscripts (λ, λ − ν) and (κ, κ − ν) stand for (A1, L), (L, S1) or (S1, S2), the following equation are obtained from equation (25): ων 1 1 dN{2p} (t ) 1 0 + N{2p} (t ) − {2p} n¯ = dt T1 ω0 N 16π " 1 × ! EA1 (t )EL∗ (t )Q∗A1 ,L;{2p} (t ) + c.c. √ μA1 LA1 μL LL 1 ! EL (t )ES∗1 (t )Q∗L,S1 ;{2p} (t ) + c.c. +√ μL LL μS1 LS1 # 1 ∗ ∗ ! + ! ES1 (t )ES2 (t )QL,S1 ;{2p} (t ) + c.c. , μS1 LS1 μS2 LS2 (39) where according to the definition (26) the order of subscripts in amplitudes of collective vibrations is arbitrary as well. For instance, when λ and λ − ν stand for S1 and S2, respectively, and κ and κ − ν stand for A1 and L, respectively, then QA1 ,L;λ,λ−ν;κ,κ−ν (t ) = QA1 ,L;S1 ,S1 ;A1 ,L (t ) = QA1 ,L;A1 ,L;S1 ,S1 (t ), QL,S1 ;λ,λ−ν;κ,κ−ν (t ) = QL,S1 ;S1 ,S1 ;A1 ,L (t ) = QA1 ,L;L,S1 ;S1 ,S1 (t ), and QS1 ,S2 ;λ,λ−ν;κ,κ−ν (t ) = QS1 ,S2 ;S1 ,S2 ;A1 ,L (t ) = QA1 ,L;S1 ,S2 ;S1 ,S2 (t ). From equation (27) we obtain the equation for the amplitude of collective vibrations Q{3p} (t ) (standing for Qα,β;γ ,δ;ε,ξ (t )), where the subscripts (α, β), (γ , δ), and (ε, ζ ) stand for (A1, L), (L, S1) or (S1, S2):

ω0 2 1 1 (Rm) gL μA1 μL 0 dQ{3p} (t ) = L c dt ων T2 8π ωL LA1 LL A1 ,L;{3p}

gS1 μL μS1 0 + EL (t )ES∗1 (t )L,S 1 ;{3p} ωS1 LL LS1

gS2 μS1 μS2 ∗ 0 + ES (t )ES2 (t )S1 ,S2 ;{3p} ωS2 LS1 LS2 1

i 2 1 − ω0 − ων2 Q{3p} (t ) − Q{3p} (t ), (40) 2ων T2 where according to the definition (28) the order of subscripts in overlap integrals is arbitrary and the noise terms are omitted. The overlap integrals on the right-hand sides of equations (38)–(40) are presented in the appendix (see equations (A.7), (A.8)). In the case where the Rayleigh length of the beams (Rm) zR,λ exceeds considerably the Raman medium length, the beam radii could be replaced by their averages along the Raman medium, rA1 (L, S1 , S2 ), and the overlap integrals on the right-hand sides of equations (38)–(40) are transformed 0 to the two-term products, ... ≈ σ...0 δ... , where all σ...0 depend only on the beam radii, and the so-called multiwave mixing 7

S V Voitikov et al


× A01 ,L;A1 ,L − 2NA1 ,L;A1 ,L (t ) √ √ gS1 μA1 μL μS1 ! ! EL (t )2 ES∗1 (t ) + ωS1 LA1 LL LS1 √ 0 gS2 μA1 μL μS1 μS2 ! × A1 ,L;L,S1 − 2NA1 ,L;L,S1 (t ) + ωS2 LA1 LL LS1 LS2 0 ∗ × EL (t )ES1 (t )ES2 (t ) A1 ,L;S1 ,S2 − 2NA1 ,L;S1 ,S2 (t ) dEA1 (t ) dEA1 (t ) + , (41) + dt sa dt loss

0 S1 ,S2 ;S1 ,S2 − 2NS1 ,S2 ;S1 ,S2 (t ) dES1 (t ) dES1 (t ) + , + dt sa dt loss

dEL (t ) dEL (t ) dEL (t ) + + , dt am dt sa dt loss

(43)

and ωS2 ω0 2 (Rm) 1 1 dES2 (t ) = c L dt 16π ων 1 + iωT2 μS1 μS2 " √ gL μA1 μL μS1 μS2 ∗ ! × E (t )EL (t )ES1 (t ) ωL LA1 LL LS1 LS2 A1 × A01 ,L;S1 ,S2 − 2NA1 ,L;S1 ,S2 (t ) √ √ μL μS1 μS2 ∗ gS ! E (t )ES1 (t )ES1 (t ) + 1 √ ωS1 LL × LS1 LS2 L 0 × L,S − 2NL,S1 ;S1 ,S2 (t ) 1 ;S1 ,S2 gS μS μS + 2 1 2 ES∗1 (t )ES1 (t )ES2 (t )S01 ,S2 ;S1 ,S2 ωS2 LS1 LS2 × S01 ,S2 ;S1 ,S2 − 2NS1 ,S2 ;S1 ,S2 (t ) dES2 (t ) dES2 (t ) + , (44) + dt sa dt loss where ω = ων − ω0 . On the right-hand sides of the equations for the populations of excited states (equation (39)), quasi-stationary solutions for the amplitudes of collective vibrations (equation (40)) should be set. Equations (41)–(44) make visible the contributions of cascade twophoton processes and processes multiwave mixing in SRSconversion. We will demonstrate it on the example of equation (42) for the amplitude of a fundamental wave. On the right-hand side of equation (42), two terms proportional to EA∗1 (t )EA1 (t )EL (t ) and EL (t )ES∗1 (t )ES1 (t ) describe SRSconversion of the first anti-Stokes radiation into fundamental radiation and of the fundamental radiation into first Stokes radiation via two-photon processes if there is no inversion of Raman medium, namely, if A01 ,L;A1 ,L > 2NA1 ,L;A1 ,L (t ) 0 and L,S > 2NL,S1 ;L,S1 (t ). Otherwise, when Raman1 ;L,S1 medium inversion takes place, these two terms continue describing SRS-conversion as two-photon processes, but due to positive inversion the first Stokes radiation is converted into fundamental radiation and the fundamental radiation is converted into first anti-Stokes radiation. The rest of the three terms on the right-hand side of equation (42), which are proportional to EA1 (t )ES∗1 (t )ES2 (t ), ES1 (t )ES1 (t )ES∗2 (t ), and EA1 EL∗ (t )ES1 (t ) describe generate and non-generate four-wave mixing. Interpretation of the corresponding terms in equations (41), (43), (44) is similar. Equations (41)–(44) contain, as special cases, the results obtained earlier. If the excitation of the Raman medium is negligibly small and only fundamental, first anti-Stokes, and first Stokes radiations are generated, then from equations (41)– (43) we obtain the equations proposed in [19–21], in which, of the possible mechanisms of SRS-conversion, only multiwave mixing is considered. If the fundamental wave is SRSconverted into only first Stokes wave without excitation of the Raman medium (NL,S1 ;L,S1 (t ) = 0), then in this particular case from equations (42), (43) we arrive at the semiclassical model of quasi-stationary intracavity SRS-conversion developed and

ωL ω0 2 (Rm) dEL (t ) = c L dt 16π ων gL μA1 μL ∗ 1 1 × E (t )EA1 (t )EL (t ) 1 + iωT2 μA1 μL ωL LA1 LL A1 √ gS μA1 μL μS1 μS2 × A01 ,L;A1 ,L − 2NA1 ,L;A1 ,L (t ) + 2 ! ωS2 LA1 LL LS1 LS2 × EA1 (t )ES∗1 (t )ES2 (t ) A01 ,L;S1 ,S2 − 2NA1 ,L;S1 ,S2 (t ) gS1 μL μS1 1 1 EL (t )ES∗1 (t )ES1 (t ) − 1 − iωT2 μL μS1 ωS1 LL LS1 √ √ gS2 μL μS1 μS2 0 ! × L,S1 ;L,S1 − 2NL,S1 ;L,S1 (t ) + √ ωS2 LL LS1 LS2 0 × ES1 (t )ES1 (t )ES∗2 (t ) L,S − 2NL,S1 ;S1 ,S2 (t ) 1 ;S1 ,S2 √ √ μA1 μL μS1 gS1 gL iωT2 ! ! + − 1 + (ωT2 )2 μA1 μL ωS1 μL μS1 ωL LA LL LS1 # 1 × EA1 EL∗ (t )ES1 (t ) A01 ,L;L,S1 − 2NA1 ,L;L,S1 (t ) +

(42)

1 dES1 (t ) ωS1 ω0 2 (Rm) 1 = cL dt 16π ων 1 + iωT2 μL μS1 √ √ gL μA1 μL μS1 ∗ ! ! × E (t )EL (t )EL (t ) ωL LA1 LL LS1 A1 × A01 ,L;L,S1 − 2NA1 ,L;L,S1 (t ) gS μL μS1 ∗ E (t )EL (t )ES1 (t ) + 1 ωS1 LL LS1 L 0 × L,S − 2NL,S1 ;L,S1 (t ) 1 ;L,S1 gS2 gS1 iωT2 + − μS1 μS2 ωS1 1 + (ωT2 )2 μL μS1 ωS2 √ √ μL μS1 μS2 ! ×√ EL (t )ES∗1 (t )ES2 (t ) LL LS1 LS2 0 × L,S − 2NL,S1 ;S1 ,S2 (t ) 1 ;S1 ,S2

gL μA1 μL μS1 μS2 1 1 − 1 − iωT2 μS1 μS2 ωL LA1 LL LS1 LS2 ×EA1 (t )EL∗ (t )ES2 (t ) A01 ,L;S1 ,S2 − 2NA1 ,L;S1 ,S2 (t ) gS μS μS + 2 1 2 ES1 (t )ES∗2 (t )ES2 (t ) ωS2 LS1 LS2 8

S V Voitikov et al


did not fit the experimental data [2]. In each laser element the beam radii were estimated by the ABCD-method using the experimental measurements of the beam waist diameters of the Nd:LSB/Cr:YAG microchip laser [33] made from the same crystals Nd:LSB and Cr:YAG as those used in [1, 2]. The description of the terms dE... (t )/dt|am and dE... (t )/dt|sa for this laser is based on the 4-multiplet spectroscopic scheme of Nd ions in the LSB-host [2, 15, 33] and the 4-level spectroscopic scheme of Cr ions in YAG-garnet [34, 35]: dEA1 (L,S1 ,S2 ) (t ) dEA1 (L,S1 ,S2 ) (t ) + dt dt

2 3 4

1 5 6 7 8

sa

(3) the attenuator, (5) the active crystal (Nd:LSB), (6) the Raman-active crystal (Ba(N)3)2), (7) the saturable absorber (Cr:YAG), and (8) the output mirror.

studied in [18, 19]. In these studies, a good agreement between the model of intracavity SRS and experimental data on the quasi-stationary intracavity generation of the first Stokes wave in hydrogen in a high-finesse-cavity Raman laser was demonstrated. If in the quasi-stationary Raman laser only a fundamental and a first Stokes wave are generated, the excitation of the Raman medium is negligibly small, and the phase relationships of waves are ignored, then on going in equations (42), (43) from amplitudes to the numbers of photon (or intensities), we obtain, as special cases, the phenomenological models of intracavity Raman conversion used in [1–13] to calculate or estimate the parameters of output Stokes radiation in compact Raman lasers. Now we will use equations (34)–(40) that take into account the first order excitation of the Raman medium to see whether the first anti-Stokes and second Stokes radiation observed in the Nd:LSB/Ba(NO3)2/Cr:YAG microchip Raman laser [1, 2] was due to the cascade SRS or/and to the transient multiwave mixing. The scheme of the laser is shown in figure 1. This laser, generating 48 ps first Stokes pulses, had a short cavity (≈0.4 cm) and, therefore, in modelling, the radii of generated and/or converted modes in the laser elements can be replaced by their average values rA1 (L,S1 ,S2 ) . According to the arrangement of laser elements (the Raman crystal was placed between the active crystal and the absorber), in the coupling factors of multiwave mixing (equations (A.9)–(A.28)) the phase incursions for the generated/converted waves between the output mirror and the crystal of barium nitrate are ϕA1 (L,S1 ,S2 ) = kA(sa) L(sa) . 1 (L,S1 ,S2 )

loss

1 = − c(σ13,A1 (L,S1 ,S2 ) − σ24,A1 (L,S1 ,S2 ) ) 2 ∞ 2 2r2 × dr2π r

2 exp − (sa)

2 0 rA1 (L,S1 ,S2 ) π rA(sa) 1 (L,S1 ,S2 ) EA (L,S ,S ) (t ) EA (L,S ,S ) (t ) 1 × na (r, t ) 1 1 2 − cσ24,A1 (L,S1 ,S2 ) na0 1 1 2 LA1 (L,S1 ,S2 ) 2 LA1 (L,S1 ,S2 ) 1 1 − c αA1 (L,S1 ,S2 ) − ln(RA1 (L,S1 ,S2 ),in RA1 (L,S1 ,S2 ),out ) 2 2 EA1 (L,S1 ,S2 ) (t ) , (46) × LA1 (L,S1 ,S2 ) ∞ 1 2 2r2 dEL (t ) = cσam 2π r

2 exp − (am) 2 dt am 2 0 rL π rL(am) EL (t ) × [nu (r, t ) − nl (r, t )] dr LL ∞ 2 − βsp (2π ωL )1/2 2π r (am) 2 0 π rL 2 nu (r, t ) 2r × exp − dr, (47) (am) 2 τul r

Figure 1. Laser scheme. (1) The laser diode, (2) and (4) the lenses,

L

where σam is the spectroscopic stimulated emission crosssection of activator atoms in the active medium, nu (r, t ) and nl (r, t ) are 2D-densities of activator atoms (the number of atoms in the active medium per unit area of the crosssection) at the upper and lower laser levels in the upper and lower laser multiplets, respectively, σ13,A1 (L,S1 ,S2 ) and σ24,A1 (L,S1 ,S2 ) are the ground- and the excited-state crosssections of the first anti-Stokes (fundamental, first Stokes, second Stokes) wave absorption in the saturable absorber, respectively, rA(am) and rA(sa) are the averaged radii of 1 (L,S1 ,S2 ) 1 (L,S1 ,S2 ) the first anti-Stokes (fundamental, first Stokes, second Stokes) beam in the active medium and the absorber, respectively, na (r, t ) is the 2D-density of absorbing atoms in the ground state in the absorber, na0 = Na L(sa) , Na is the 3D-density of absorbing atoms in the absorber, L(sa) is the saturable absorber length, RA1 (L,S1 ,S2 ),in and RA1 (L,S1 ,S2 ),out are the reflection coefficients of the rear and output mirrors at the first antiStokes (fundamental, first Stokes, second Stokes) wavelength, respectively, αA1 (L,S1 ,S2 ) is the internal loss at the first antiStokes (fundamental, first Stokes, first Stokes) wavelength, βsp is the fraction of spontaneously emitted photons contributing to the fundamental wave. The rate equations for the Nd:LSB active medium were presented in [2, 15, 33]. We denote by num (r, t ) and nlm (r, t )

(45)

It is typical of compact (short-cavity) pulsed Raman lasers that the inhomogeneity of transverse distributions of beam intensities is significant and, respectively, the beam radii are small (see, e.g., [2, 15–17]). In [2], the comparison of the obtained experimental data with the results of calculations based on the model taking into account transverse distribution of beam intensities and with the results of calculations based on 1D models (plane wave approximation) was presented. The model taking into account transverse distribution of beam intensities demonstrated good agreement with the experimental data while the 1D model 9

S V Voitikov et al


the 2D-densities (the number of atoms in the active medium per unit area of the cross-section) of activator atoms in the upper and lower laser multiplets, respectively, and denote by nuu (r, t ) and nll (r, t ) the 2D-densities of activator atoms at all levels of the upper and lower multiplets except for the laser levels, respectively (thus, num (r, t ) = nu (r, t ) + nuu (r, t ) and nlm (r, t ) = nl (r, t ) + nll (r, t )). The rate equations for the active medium are [2, 15, 33] 2 dnu (r, t ) 1 nu (r, t ) 2 2r =− + ηu ηeff exp − 2 WP (am) 2 dt τul ω r P π rP P 2 2r2 − cσam [nu (r, t ) − nl (r, t )]

2 exp − (am) 2 rL π rL(am) ∗ E (t )EL (t ) nu (r, t ) − fu num (r, t ) × L − , (48) 8π ωL LL τu

Figure 2. Experimental (dots, after [2] (© 2006 Elsevier)) and calculated (solid line) shapes of the first Stokes pulse in Nd:LSB/Ba(NO3)2/Cr:YAG Raman laser.

1 2 2r2 dnuu (r, t ) = (1 − ηu )ηeff exp − WP 2 (am) 2 dt ω r P π r P

×

nuu (r, t ) − (1 − fu )num (r, t ) , τu

(49)

dnl (r, t ) 2 = cσam [nu (r, t ) − nl (r, t )] $ %2 dt (am) π rL ⎞ ⎛ 2r2 ⎟ EL∗ (t )EL (t ) ⎜ × exp ⎝− $ %2 ⎠ 8π ωL LL rL(am) −

nl (r, t ) − fl nlm (r, t ) nu (r, t ) + , τl τul

(50)

dnll (r, t ) nll (r, t ) − (1 − fl )nlm (r, t ) nll (r, t ) − , =− dt τl τlg

(51)

where ηu is the fraction of nonradiative transitions of activator atoms from the upper pump bands to the upper laser level, τul is the upper laser level lifetime of activator atoms, τu and τl are the thermalization times of activator atoms in the upper and lower laser multiplets, respectively, τlg is the activator atom lifetime in the lower laser multiplet, fu and fl are the Boltzmann factors of the upper and lower laser levels, respectively, ηeff is the pump absorption efficiency, ωP and WP are the pump photon energy and the pump power, respectively, and rP(am) is the effective Gaussian radius of the pump beam in the active medium. The absorber saturation and recovery dynamics is described from the viewpoint of the 4-level scheme of chromium ions and takes into account the ground- and excitedstate absorption. For the 2D-density (the number of atoms in the absorber per unit area of the cross-section) of chromium ions in the ground state na (r, t ) the rate equation is of the form [34, 35] dna (r, t ) 2 2r2 = −cna (r, t ) σ13,L

2 exp − (sa) 2 dt rL π rL(sa) ∗ 2 2r2 EL (t )EL (t ) + σ13,S1 ×

2 exp − (sa) 2 8π ωL LL r π r (sa) S1

2

2r2

+ σ13,S2

2 exp − (sa) 2 π rS(sa) r 2 S2 ∗ ES2 (t )ES2 (t ) 2 2r2 + σ13,A1 ×

2 exp − (sa) 2 8π ωS2 LS2 rA1 π rA(sa) 1 ∗ EA (t )EA1 (t ) na0 − na (r, t ) × 1 − , (52) 8π ωA1 LA1 τ21 where τ21 is the excited-state lifetime of chromium ions. In modelling, we considered the case of ωv = ω0 . The laser parameters used in the calculations are presented in table 1. The energy, duration and shape of the pulses of generated/converted waves were calculated by solving equations (34)–(40), (46)–(52). In calculations, it was obtained that the peak intensity in the centre of a laser beam was less than 1250 MW cm–2, that permitted the effects of non-resonant third-order nonlinearities to be ignored. The increment of the laser beam amplification in an active crystal was calculated to be less than 0.15, while the increment of the Stokes beam amplification in a Raman-active crystal was less than 2.5. The latter is sufficiently less than 30 which is necessary to reach the SRS threshold [40, 41]. The transmittance of the saturated absorber was estimated to be of 94%–96%. This confirms the applicability of the standing waves’ approximation. The calculated duration and pulse energies were ≈0.65 ns and 1.8 μJ (fundamental pulse), 53 ps and 2.7 μJ (first Stokes pulse), 80 ps and 0.02 μJ (second Stokes pulse), ≈0.25 ns and 0.001 μJ (first anti-Stokes pulse). The calculated peak power of the first Stokes pulse was ≈49 kW, which is quite close to the experimentally measured (≈48 kW) [2]. Because the reflection coefficient of the output mirror for the first antiStokes wave was approximately five times larger than that for the first Stokes wave, the first anti-Stokes photon lifetime in a cavity was approximately five times longer than the first Stokes photon lifetime. Thus, the duration of the first antiStokes pulse was approximately five times longer than that of the first Stokes pulse. Figure 2 presents the theoretically calculated first Stokes pulse along with the experimentally recorded one [2]. Figure 2 demonstrates a good agreement between the calculated and the experimental shapes of the first Stokes pulse, except for

P

−

ES∗1 (t )ES1 (t )

S1

10

8π ωS1 LS1

S V Voitikov et al


Table 1. Parameters of laser elements.

Designation

Symbol

Value

Fundamental (laser) wavelength of Nd:LSB crystal, nm First Stokes wavelength, nm Second Stokes wavelength, nm First anti-Stokes wavelength, nm Nd:LSB spectroscopic stimulated radiation cross-section, sm2 Nd:LSB-crystal length, mm Refractive index of Nd:LSB crystal at λA1 Refractive index of Nd:LSB crystal at λL Refractive index of Nd:LSB crystal at λS1 Refractive index of Nd:LSB crystal at λS2 Ba(NO3)2-crystal length, mm Raman shift of Ba(NO3)2 crystal, cm−1 Raman gain of Ba(NO3)2 crystal at λL , cm GW–1 Raman gain of Ba(NO3)2 crystal at λS1 , cm GW–1 Raman gain of Ba(NO3)2 crystal at λS2 , cm GW–1 Refractive index of Ba(NO3)2 crystal at λA1 Refractive index of Ba(NO3)2 crystal at λL Refractive index of Ba(NO3)2 crystal at λS1 Refractive index of Ba(NO3)2 crystal at λS2 Dephasing time of Ba(NO3)2 crystal, ps Relaxation time of Ba(NO3)2 crystal, ns Refractive index of YAG crystal at λA1 Refractive index of YAG crystal at λL Refractive index of YAG crystal at λS1 Refractive index of YAG crystal at λS2 Initial transmittance of absorber, % Cr4+:YAG saturable absorber length, mm Effective radius of pump beam, μm Reflection coefficients of rear mirror, % Internal losses at λL , λS1 , λS2 , and λA1 Reflection coefficient of output mirror at λL , % Reflection coefficient of output mirror at λS1 , % Reflection coefficient of output mirror at λS2 , % Reflection coefficient of output mirror at λA1 , % Radius of fundamental beam in Raman medium, μm Radius of first Stokes beam in Raman medium, μm Radius of second Stokes beam in Raman medium, μm Radius of first anti-Stokes beam in Raman medium, μm Ground-state absorption cross-section of Cr4+ ions at λL , cm2 Excited-state absorption cross-section of Cr4+ ions at λL , cm2 Ground-state absorption cross-section of Cr4+ ions at λS1 , cm2 Exited-state absorption cross-section of Cr4+ ions at λS1 , cm2 Ground-state absorption cross-section of Cr4+ ions at λS2 , cm2 Exited-state absorption cross-section of Cr4+ ions at λS2 , cm2 Ground-state absorption cross-section of Cr4+ ions at λA1 , cm2 Exited-state absorption cross-section of Cr4+ ions at λA1 , cm2 Excited-state lifetime of Cr4+ ions, μs

λL λS1 λS2 λA1 σam L(am) μam,A1 μam,L μam,S1 μam,S2 L(Rm) ω0 gL gS1 gS2 μA1 μL μS1 μS2 T2 T1 μsa,A1 μsa,L μsa,S1 μsa,S2 Tini L(sa) rP(am) RL,in ≈ RS1 ,in ≈RS2 ,in ≈ RA1 ,in αL,in ≈ αS1 ,in ≈ αS2 ,in ≈ αA1 ,in RL,in RS1 ,in RS2 ,in RA1 ,in rL(Rm) ≈ rL(am) ≈ rL(sa) ≈ rL ≈ rS(am) ≈ rS(sa) ≈ rS1 rS(Rm) 1 1 1 (Rm) (am) ≈ rS2 rS2 ≈ rS2 ≈ rS(sa) 2 (am) (sa) ≈ r ≈ r ≈ r A1 rA(Rm) A1 A1 1 σ13,L σ24,L σ13,S1 σ24,S1 σ13,S2 σ24,S2 σ13,A1 σ24,A1 τ21

1063 [36] 1196 1367 957 2.1 10−19 [36] 1.24 1.8299 [36] 1.8269 [36] 1.8242 [36] 1.8217 [36] 2 1047 13.5 [37] 10. [37] 9.5 [37] 1.5569 [38] 1.555 [38] 1.5532 [38] 1.5512 [38] 25 0.5 1.8184 [39] 1.8169 [39] 1.8155 [39] 1.8144 [39] 78 0.86 ≈40 99.94–99.98 ≈0.02 97 16 32 96 41 36 31 38 3.5 10−18 5. 10−19 0.2σ13,L 0.2σ24,L 0.05σ13,L 0.05σ24,L 0.9σ13,L 0.9σ24,L 4

the trailing edge of the pulse, where in the experiment a pulse repetition was observed. In [2] the energy of the second Stokes pulse was not measured or estimated (as well as that of the first anti-Stokes pulse), but the calculation fulfilled here shows it to be rather small. Thus, it is unlikely that the generation of the second Stokes radiation could cause the pulse repetition in the oscilloscope trace in figure 2 as it was earlier suggested in [2]. We think the reason for such pulse repetition is probably the generation of high-order transverse modes of fundamental radiation, whose initiation starts when the loss of the fundamental mode TEM00 begins to increase rapidly because of the intense SRS. The subsequent SRSconversion of the higher-order transverse fundamental modes

into Stokes radiation caused, in all likelihood, pulse repetition at the trailing edge of the Stokes pulse. In the calculations, the concentration N of the Raman-active centres was assumed to be 1021 cm–3 (in condensed matter N can be from 1020 cm–3 to 1023 cm–3 [29]). At such a concentration of Raman-active centres, the maximum values of all the populations on the righthand side of equation (38) differed from the equilibrium values very insignificantly. Accordingly, the generation of first antiStokes pulses was mainly due to the multiwave mixing without cascade two-photon Raman scattering. This is confirmed both experimentally in [1], where first anti-Stokes radiation was registered only in the spectrum, and by the present calculations, according to which the energy of the first anti-Stokes pulses 11

S V Voitikov et al


δχλ,λ−ν;κ,κ−ν (z) kλ kλ−ν kκ kκ−ν = − − + , 2Rλ (z) 2Rλ−ν (z) 2Rκ (z) 2Rκ−ν (z) δwλ,λ−ν;κ,κ−ν (z) = arcsin δχλ,λ−ν;κ,κ−ν (z)

is really very small. The calculated energy of the second Stokes pulse is 20 times larger than the calculated energy of the first anti-Stokes pulse, and the contribution of the twophoton (cascade) processes to the generation of the second Stokes radiation exceeds insignificantly the contribution of the multiwave mixing. Thus, from the results of calculations it follows that the problem of obtaining first anti-Stokes pulses in compact passively Q-switched Raman lasers as a result of cascade twophoton SRS at the moment still remains unsolved. The most obvious condition for obtaining such pulses, as we see it, is increasing the intracavity intensity of fundamental and Stokes pulses and optimizing their duration in order to increase the population of the excited states of the Raman medium. One possible solution of this problem is the use of the scheme of external pumping by high-power nanosecond pulses.

−1 1 1 1 1 . (A.6) + + + rλ (z)2 rλ−ν (z)2 rκ (z)2 rκ−ν (z)2 After integration in expression (23) with respect to the 0 transverse radius r the overlap integrals A01 ,L;A1 ,L , L,S , 1 ;L,S1 0 and S1 ,S2 ;S1 ,S2 become 2 1 0 λ,λ−ν;λ,λ−ν = [rλ (z)2 + rλ−ν (z)2 ]−1 dz, π L(Rm) L(Rm) (A.7)

×

where the set of subscripts (λ, λ − ν; λ, λ − ν) stands for (A1 , L; A1 , L), (L, S1 ; L, S1 ) or (S1 , S2 ; S1 , S2 ), respectively, 0 and the overlap integrals A01 ,L;L,S1 , L,S , and A01 ,L;S1 ,S2 1 ;S1 ,S2 read , 1 1 0 = π2 L(Rm) λ,λ−ν;κ,κ−ν L(Rm) r (z)r (z)r λ λ−ν κ (z)rκ−ν (z)

4. Conclusion On the basis of the semiclassical approach in describing the interaction between the field and a nonlinear medium we propose a semiclassical theory of transient intracavity SRS-conversion in a laser generating multiple Stokes and/or anti-Stokes waves with excitation of the Raman medium. From the wave equation for the field in the cavity in the slowly varying amplitudes, rotating wave, and paraxial approximations a hierarchy of equations for the amplitudes of generated/converted waves, the amplitudes of collective vibrations of the Raman medium, and the populations of excited states of the Raman medium has been obtained. It has been shown that the known semiclassical and phenomenological models of intracavity SRS-conversion are special cases of the proposed theory. We used the theory to calculate the output pulse parameters of a solid-state Raman microchip laser in which we obtained experimentally, along with high peak power first Stokes pulses, also first anti-Stokes and second Stokes radiation but only in the spectrum. The results of modelling agree well with the experimental data and show that anti-Stokes radiation is due to the multiwave mixing.

cos[δkλ,λ−ν;κ,κ−ν z+δψλ,λ−ν;κ,κ−ν (z)+δϕλ,λ−ν;κ,κ−ν +δwλ,λ−ν;κ,κ−ν (z)]

+r

1 2 λ−ν (z)

0 σλ,λ−ν;λ,λ−ν =

1 1 + rκ (z) 2 +r 2 ) κ−ν (z)

2

, 2 2 π rλ + rλ−ν

δλ,λ−ν;λ,λ−ν = 1,

2

1/2

d z,

(A.9) (A.10)

for (λ, λ − ν; κ, κ − ν) = (A1 , L; L, S1 ), (L, S1 ; S1 , S2 ) or (A1 , L; S1 , S2 ) −1 4 1 1 1 1 0 σλ,λ−ν;κ,κ−ν = + 2 + 2 + 2 , π rλ rλ−ν rκ rκ−ν rλ2 rκ rλ−ν rκ−ν (A.11) δλ,λ−ν;κ,κ−ν

(A.1)

=

sin(δkλ,λ−ν;κ,κ−ν L(Rm) +δϕλ,λ−ν;κ,κ−ν )−sin(δϕλ,λ−ν;κ,κ−ν ) , 2δkλ,λ−ν;κ,κ−ν L(Rm)

(A.12)

δψλ,λ−ν;κ,κ−ν (z) = arcsin

1 2 λ (z)

where the set of subscripts (λ, λ − ν; λ, λ − ν) stands for (A1 , L; L, S1 ), (L, S1 ; S1 , S2 ) or (A1 , L; S1 , S2 ), respectively. (Rm) When the Rayleigh length of the beams zR,λ sufficiently exceeds the Raman medium length, the beam radii along the Raman medium are almost constant, rA(Rm) (z) ≈ 1 (L,S1 ,S2 ) rA1 (L,S1 ,S2 ) , and the overlap integrals on the right-hand sides of equations (38)–(42) can be given as two-term products, 0 ... ≈ σ...0 δ... , where for (λ, λ − ν; λ, λ − ν) = (A1 , L; A1 , L), (L, S1 ; L, S1 ) or (S1 , S2 ; S1 , S2 )

To make the final expressions for overlap integrals more compact, the following designation were introduced z − z0λ , zRλ

(δχλ,λ−ν;κ,κ−ν (z))2 + ( r

(A.8)

Appendix

cλ (z) =

(A.5)

δkλ,λ−ν;κ,κ−ν = (kλ − kλ−ν − kκ + kκ−ν ),

(A.3)

for (λ, λ − ν; λ, λ − ν; λ, λ − ν; λ, λ − ν) = (A1 , L; A1 , L; A1 , L; A1 , L), (L, S1 ; L, S1 ; L, S1 ; L, S1 ) or (S1 , S2 ; S1 , S2 ; S1 , S2 ; S1 , S2 ) −1 1 9 1 0 σλ,λ−ν;λ,λ−ν;λ,λ−ν;λ,λ−ν = + 2 (A.13) 2 4π 3 rλ2 rλ−ν rλ2 rλ−ν

δϕλ,λ−ν;κ,κ−ν = (ϕλ − ϕλ−ν − ϕκ + ϕκ−ν ),

(A.4)

δλ,λ−ν,λ,λ−ν,λ,λ−ν,λ,λ−ν = 1,

×

(1−cλ (z)cκ−ν (z))(cλ−ν (z)+cκ (z))−(1−cλ−ν (z)cκ (z))(cλ (z)+cκ−ν (z))

√

(1+cλ (z)2 )(1+cλ−ν (z)2 )(1+cκ (z)2 )(1+cκ−ν (z)2 )

,

(A.2)

12

(A.14)

S V Voitikov et al


for (λ, λ − ν; λ, λ − ν; λ − ν, λ − 2ν; λ − ν, λ − 2ν) = (A1 , L; A1 , L; L, S1 ; L, S1 ) or (L, S1 ; L, S1 ; S1 , S2 ; S1 , S2 ) 0 σλ,λ−ν;λ,λ−ν;λ−ν,λ−2ν;λ−ν,λ−2ν

=

3 4 r2 π 3 rλ2 rλ−ν λ−2ν

1 2 1 + 2 + 2 rλ2 rλ−ν rλ−2ν

δA1 ,L;L,S1 ;S1 ,S2 ;S1 ,S2 =

−1 ,

+ 14

(A.15)

δλ,λ−ν;λ,λ−ν;λ−ν,λ−2ν;λ−ν,λ−2ν = 1,

(Rm) +(ϕL −2ϕS1 +ϕS2 )]−sin(ϕL −2ϕS1 +ϕS2 ) 3 sin[(kL −2kS1 +kS2 )L 4 2(kL −2kS1 +kS2 )L(Rm)

sin[(kA1 −3kS1 +2kS2 )L(Rm) +(ϕA1 −3ϕS1 +2ϕS2 )]−sin(ϕA1 −3ϕS1 +2ϕS2 ) , 2(kA1 −3kS1 +2kS2 )L(Rm)

(A.26)

(A.16) σA01 ,L;L,S1 ;L,S1 ;S1 ,S2 =

σA01 ,L;A1 ,L;S1 ,S2 ;S1 ,S2

2 = 3 2 2 2 2 π rA1 rL rS1 rS2

1 1 1 1 + 2 + 2 + 2 rA2 1 rL rS1 rS2

−1

,

δA1 ,L;A1 ,L;S1 ,S2 ;S1 ,S2 = 1,

×

(A.17)

×

3 3 1 1 + 2 + 2 + 2 2 rA1 rL rS1 rS2

−1 ,

(A.19)

9 π 3 rA1 rL rS31 rS32 −1 1 1 3 3 × + 2 + 2 + 2 , rA2 1 rL rS1 rS2

[1] Demidovich A A et al 2003 Appl. Phys. B 76 509 [2] Demidovich A A, Voitikov S V, Batay L E, Grabtchikov A S, Danailov M B, Lisinetskii V A, Kuzmin A N and Orlovich V A 2006 Opt. Commun. 263 52 [3] Pearce S, Ireland C L M and Dyer P E 2006 Opt. Commun. 260 680 ˇ y P, Zendizan ˙ ˇ [4] Cern` W, Jabczyński J, Jel´ınková H, Sulc J and Kopczyński K 2002 Opt. Commun. 209 403 ˇ y P, Jel´ınková H, Zverev P G and Basiev T T 2004 Prog. [5] Cern` Quantum Electron. 28 113 [6] Hamano A, Pleasants S, Okida M, Itoh M, Yatagai T, Watanabe T, Fujii M, Iketaki Y, Yamamoto K and Omatsu T 2006 Opt. Commun. 260 675 [7] Ding S, Zhang X, Wang Q, Su F, Jia P, Li S, Fan S, Chang J, Zhang S and Liu Z 2006 IEEE J. Quantum Electron. 42 927 [8] Ding S, Zhang X, Wang Q, Su F, Li S, Fan S, Zhang S, Chang J, Wang S and Liu Y 2006 Appl. Phys. B 85 89 [9] Chen W, Inagawa Y, Omatsu T, Tateda M, Takeuchi N and Usuki Y 2001 Opt. Commun. 194 401 [10] Chen Y F, Su K W, Zhang H J, Wang J Y and Jiang M H 2005 Opt. Lett. 30 3335 [11] Chang Y T, Su K W, Chang H L and Chen Y F 2009 Opt. Express 17 4330 [12] Su F, Zhang X, Wang Q, Jia P, Li S, Liu B, Zhang X, Cong Z and Wu F 2007 Opt. Commun. 277 379 [13] Piper J A and Pask H M 2007 IEEE J. Select. Top. Quantum Electron. 13 692 [14] Sabella A, Piper J A and Mildren R P 2010 Opt. Lett. 35 3874 [15] Voitikov S V, Demidovich A A, Shpak P V, Grabtchikov A S, Danailov M B and Orlovich V A 2010 J. Opt. Soc. Am. B 27 1232 [16] Shpak P V, Voitikov S V, Demidovich A A, Danailov M B and Orlovich V A 2012 Appl. Phys. B 108 269 [17] Shpak P V, Voitikov S V, Chulkov R V, Apanasevich P A, Orlovich V A, Grabtchikov A S, Kushwaha A, Satti N, Agrawal L and Maini A K 2012 Opt. Commun. 285 3659 [18] Brasseur J K, Roos P A, Repasky K S and Carlsten J L 1999 J. Opt. Soc. Am. B 16 1305

(A.21)

8 π 3 rA2 1 rL3 rS21 rS2 −1 2 3 2 1 × + 2 + 2 + 2 , rA2 1 rL rS1 rS2

σA01 ,L;A1 ,L;L,S1 ;S1 ,S2 =

(A.23)

δA1 ,L;A1 ,L;L,S1 ;S1 ,S2 (Rm) +(2ϕA1 −3ϕL +ϕS2 )]−sin(2ϕA1 −3ϕL +ϕS2 ) 1 sin[(2kA1 −3kL +kS2 )L 4 2(2kλ −3kλ−ν +kκ−ν )L(Rm) sin[(kL −2kS1 +kS2 )L(Rm) +(ϕL −2ϕS1 +ϕS2 )]−sin(ϕL −2ϕS1 +ϕS2 ) + 34 , 2(kL −2kS1 +kS2 )L(Rm)

=

(A.24)

×

(A.27)

References

δA1 ,L;S1 ,S2 ;S1 ,S2 ;S1 ,S2 sin(δkA1 ,L;S1 ,S2 L(Rm) + δϕA1 ,L;S1 ,S2 ) − sin(δϕA1 ,L;S1 ,S2 ) = , 2δkA1 ,L;S1 ,S2 L(Rm) (A.22)

,

(A.28)

σA01 ,L;S1 ,S2 ;S1 ,S2 ;S1 ,S2 =

π 3r

−1

9 10 +(ϕA1 −ϕL −ϕS1 +ϕS2 )]−sin(ϕA1 −ϕL −ϕS1 +ϕS2 ) sin[(kA1 −kL −kS1 +kS2 )L 2(kA1 −kL −kS1 +kS2 )L(Rm) 1 + 10 sin[(kA1−3kL+3kS1 −kS2 )L(Rm) +(ϕA1 −3ϕL +3ϕS1 −ϕS2 )]−sin(ϕA1 −3ϕL +3ϕS1 −ϕS2 ) . 2(kA1 −3kL +3kS1 −kS2 )L(Rm)

δA1 ,L;A1 ,L;A1 ,L;S1 ,S2 sin(δkA1 ,L;S1 ,S2 L(Rm) + δϕA1 ,L;S1 ,S2 ) − sin(δϕA1 ,L;S1 ,S2 ) = , 2δkA1 ,L;S1 ,S2 L(Rm) (A.20)

σA01 ,L;L,S1 ;S1 ,S2 ;S1 ,S2 =

1 3 3 1 + 2 + 2 + 2 rA2 1 rL rS1 rS2

(Rm)

π 3 rA3 1 rL3 rS1 rS2

10 3 3 A1 rL rS1 rS2

δA1 ,L;L,S1 ;L,S1 ;S1 ,S2 =

(A.18) 9

σA01 ,L;A1 ,L;A1 ,L;S1 ,S2 =

π 3r

8 2 3 2 A1 rL rS1 rS2

1 2 3 2 + 2 + 2 + 2 2 rA1 rL rS1 rS2

−1 ,

(A.25)

13

S V Voitikov et al


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