Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field

Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field

On the basis of the general Lamb model the set of six coupled nonlinear differential equations has been derived for two-mode l = 0.63 mm laser operation with the presence both amplitude and phase anisotropy and axial magnetic field. Numeric integration of the set of equations and the Lyapunov stabil...

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Datum:1999
Hauptverfasser: Kononchuk, G.L., Yegorov, S.M.
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Veröffentlicht: Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України 1999
Schriftenreihe:Semiconductor Physics Quantum Electronics & Optoelectronics
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/119111
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spelling irk-123456789-1191112017-06-05T03:02:27Z Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field Kononchuk, G.L. Yegorov, S.M. On the basis of the general Lamb model the set of six coupled nonlinear differential equations has been derived for two-mode l = 0.63 mm laser operation with the presence both amplitude and phase anisotropy and axial magnetic field. Numeric integration of the set of equations and the Lyapunov stability analysis have been proceeded. It turned out that in zero magnetic field in the presence of the amplitude anisotropy both stable time-independed states with parallelly or orthogonally polarized modes and non-stationary state are possible. Changing orientation and nonorthogonality/nonparallelity of polarization planes in magnetic field are considered. Influence of mode-mode interaction is discussed. 1999 Article Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field / G.L. Kononchuk, S.M. Yegorov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 2. — С. 36-41. — Бібліогр.: 3 назв. — англ. 1560-8034 PACS 42.55.Lt http://dspace.nbuv.gov.ua/handle/123456789/119111 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description On the basis of the general Lamb model the set of six coupled nonlinear differential equations has been derived for two-mode l = 0.63 mm laser operation with the presence both amplitude and phase anisotropy and axial magnetic field. Numeric integration of the set of equations and the Lyapunov stability analysis have been proceeded. It turned out that in zero magnetic field in the presence of the amplitude anisotropy both stable time-independed states with parallelly or orthogonally polarized modes and non-stationary state are possible. Changing orientation and nonorthogonality/nonparallelity of polarization planes in magnetic field are considered. Influence of mode-mode interaction is discussed.
format Article
author Kononchuk, G.L.
Yegorov, S.M.
spellingShingle Kononchuk, G.L.
Yegorov, S.M.
Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
Semiconductor Physics Quantum Electronics & Optoelectronics
author_facet Kononchuk, G.L.
Yegorov, S.M.
author_sort Kononchuk, G.L.
title Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
title_short Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
title_full Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
title_fullStr Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
title_full_unstemmed Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field
title_sort polarization unstabilities in a quasi-isotropic he-ne laser in axial magnetic field
publisher Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
publishDate 1999
url http://dspace.nbuv.gov.ua/handle/123456789/119111
citation_txt Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field / G.L. Kononchuk, S.M. Yegorov // Semiconductor Physics Quantum Electronics & Optoelectronics. — 1999. — Т. 2, № 2. — С. 36-41. — Бібліогр.: 3 назв. — англ.
series Semiconductor Physics Quantum Electronics & Optoelectronics
work_keys_str_mv AT kononchukgl polarizationunstabilitiesinaquasiisotropichenelaserinaxialmagneticfield
AT yegorovsm polarizationunstabilitiesinaquasiisotropichenelaserinaxialmagneticfield
first_indexed 2025-07-08T15:14:35Z
last_indexed 2025-07-08T15:14:35Z
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fulltext 36 © 1999, Institute of Semiconductor Physics, National Academy of Sciences of Ukraine Semiconductor Physics, Quantum Electronics & Optoelectronics. 1999. V. 2, N 2. P. 36-41. 1. Introduction The roots of a quasi-isotropic He-Ne laser have been set up in the paper of Sargent et al. [1] where the general equa- tions for intensities and frequencies for an arbitrary atomic transition, any number of excited longitudinal modes and in the presence of magnetic field of arbitrary direction have been derived. But influence of a cavity anisotropy was not described consistently. Another theory has been developed in the paper of Lenstra [2]. In this work influence of differ- ent cavity anisotropy types on behaviour of polarization characteristics was considered in general for an arbitrary atomic transition of laser having been placed in axial or transversal magnetic field. Single-mode laser operating was considered on the ba- sis of the papers [1,2] for a wide range of atomic transitions and laser parameters (see, for example, [2,3] and references therein). For two-mode operation such detailed analysis has not yet been given. In the paper of Lenstra [2] qualitative remarks were made about behaviour of a j=1⇒j=2 transi- tion laser without magnetic field and in the case of weak linear phase or amplitude anisotropy. The investigation of Svirina et al. [3] is restricted to the case of a phase cavity anisotropy of a j=1⇒j=2 (λ=0.63 µm) laser in axial mag- netic field. In the present paper we have derived, within the frame- work of the general Lamb model, the two-mode operation equations with the presence both amplitude and phase anisotropy and axial magnetic field. Each spatial mode is described as a sum of two ones polarized right- and left- circularly. So, we get the set of six coupled nonlinear dif- ferential equations for four intensities of polarization modes and two phase angles of spatial modes. Numeric solution is carried out by the Runge-Kutta routine for the j=1⇒j=2 transition and with only amplitude anisotropy being pre- sented. The time-independent solutions with different po- larization states we had got were tested with respect to sta- bility by the Lyapunov analysis, and then a conclusion about existence of stationary states has been made. 2. Intensity-and phase-determining equations The present paper is based upon the formalism brought for- ward by Sargent et al. [1]. In accordance to it, a laser field is a sum of cavity eigenstates, and the active medium polari- zation is calculated by the density matrix routine up to the third terms by the electromagnetic field. Each longitudinal mode is described as a sum of two ones polarized right- and left-circularly. There are two equations for each circular mode: for its intensity and for its phase. So, there are 4n equations in the full equation set, where n is the number of longitudinal modes. However, because of all right parts of the equations include a phase difference (so called phase angle) but not phases separately, we can replace two sepa- PACS 42.55.Lt Polarization unstabilities in a quasi-isotropic He-Ne laser in axial magnetic field G.L. Kononchuk, S.M. Yegorov* Taras Shevchenko Kyiv Univ., 6 Glushkova Prosp., 252127 Kyiv, Ukraine Tel. 8 044 261 36 77, E-mail: lukich@optics.ups.kiev.ua * Tel. 8 044 443 02 93, E-mail: sequenser@mail.ru Abstract. On the basis of the general Lamb model the set of six coupled nonlinear differential equations has been derived for two-mode λ = 0.63 mm laser operation with the presence both amplitude and phase anisotropy and axial magnetic field. Numeric integration of the set of equations and the Lyapunov stability analysis have been proceeded. It turned out that in zero magnetic field in the presence of the amplitude anisotropy both stable time-independed states with parallelly or orthogonally polarized modes and non-stationary state are possible. Changing orientation and nonorthogonality/nonparallelity of po- larization planes in magnetic field are considered. Influence of mode-mode interaction is discussed. Keywords: anisotropy, He-Ne laser, magnetic field, polarization unstability. Paper received 28.05.99; revised manuscript received 08.07.99; accepted for publication 12.07.99. G.L. Kononchuk, S.M. Yegorov: Polarization unstabilities in a quasi-isotropic... 37SQO, 2(2), 1999 rate equations for phases with one for a phase angle. This is correct if we consider only time-independent solutions. Hence, the full set includes 3n equations. In the case of two longitudinal laser modes we have the following intensity- and phase-determining equations: where In+ , In– are the intensities of the right- and left-handed circular waves of the n-th spatial mode, ψn is its phase an- gle, Q c L l lxy x y= −( )( )4 is the amplitude cavity anisotropy and ~ ( )( )νxy x yc L= −4 φ φ is the phase one, c is the light velocity, L is the cavity length, lx and ly are losses, and φx and φy are phase changes for the waves polarized in x- and y-direction. α, β, θ, θ’ , σ, ρ, τ, τ’ are calculated from the density matrix motion equation. These calculations and used assumptions are adduced in the Appendix. Let’s note that in amplitude equations an appropriate α coefficient is re- sponsible for a mode gain while β and θ’s are the self-satu- ration and cross-saturation parameters, and in the phase equations σ, ρ, τ coefficients are responsible for the mode frequency pulling/pushing phenomena (change of the op- eration frequency to or from the centre of the gain line with respect to the eigenfrequency of an empty cavity). These coefficients have their analogues in single-mode equations. But θ’ , τ’ don’t have. We can say that they are responsible for some sort of gain saturation and frequency change as well but those ones depend besides on mode-mode phase relations. 2. Time-independent solutions A) Zero magnetic field Numeric integration of the set of equations (1)-(6) was per- formed for the j=1⇒j=2 (λ = 0.63 µm) transition and fol- lowing parameters: the Lorentz width γab = 225 MHz, the upper and the lower atomic levels width γa=160 MHz and γb = 290 MHz, the gain η = 1.33 ... 1.38, the Doppler pa- rameter Ku = 1010 MHz, the spatial mode interval D =ν2- ν1 = 640 MHz and several values of the amplitude anisotropy ( ) ( ) ( )11 1 1 2122 1 1 121 22 1 1 122122111111 1 1 sin~cossincos = ψνψψψτψψ θθθθβα xyxyQ I I II I I II I I IIII I I −−−′−−× ×′−−−−− + − −+ + − + −+ + − +−−++++−±+++ + + & (1) ( ) ( ) ( )11 1 1 2122 1 1 121 22 1 1 122122111111 1 1 sin~cossincos = ψνψψψτψψ θθθθβα xyxyQ I I II I I II I I IIII I I +−−′+−× ×′−−−−− − + −+ − + − −+ − + −−−−++−++−−− − − & (2) ( ) ( ) ( )22 2 2 1211 2 2 212 11 2 2 211211222222 2 2 sin~cossincos = ψνψψψτψψ θθθθβα xyxyQ I I II I I II I I IIII I I −−′−−× ×′−−−−− + − −−+ + − + −+ + − +−−++++−±+++ + + & (3) ( ) ( ) ( )22 2 2 1211 2 2 212 11 2 2 2112112212222 2 2 sin~cossincos = ψνψψψτψψ θθθθβα xyxyQ I I II I I II I I IIII I I +−−′+−× ×′−−−−− − + −+ − + − −+ − + −−−−++−++−−−− − − & (4) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         −+        ++−′+′+−× ×′−′−−−−−−−−−− + − − + + − − + −+−+ −+−+−−−−+++−++−−±+++−+ 1 1 1 1 1 1 1 1 1 121221121 22112212122121111111111 cos~sinsincos = I I I I I I I I QII IIIIII xyxy ψνψψψθθψψ ττττττρττρσσψ& (5) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )         −+        ++−′+′+−× ×′−′−−−−−−−−−− + − − + + − − + −+−+ −+−+−−−−+++−++−−±+++−+ 2 2 2 2 2 2 2 2 2 212112212 11221121211212222222222 cos~sinsincos = I I I I I I I I QII IIIIII xyxy ψνψψψθθψψ ττττττρττρσσψ& (6) G.L. Kononchuk, S.M. Yegorov: Polarization unstabilities in a quasi-isotropic... 38 SQO, 2(2), 1999 Qxy. It turned out that at weak anisotropy the state with modes polarized linearly in orthogonal planes is stable on the most part of the gain profile while the nearest to the centre mode is polarized in the lower loss plane. Hence, each mode be- ing moved through the gain profile undergoes two changes of a polarization plane. This is outlined in the Fig.1. When a mode is in a hatched area it is polarized in the lower loss plane. That is conventionally signed with a horizontal ar- row in the figure. At the same time the another mode is in a clear area and has the orthogonal polarization signed with a vertical arrow. Frequencies ω0±D/2 correspond to the sym- metric location of the modes with respect to the centre of transition line. It is just this point that has the unstable range where none of solutions is stable. Those are the dotted ar- eas in the Fig.1. When anisotropy being rised from 0 to certain value the unstability range spreads on full duty cy- cle. This value is about 10 kHz for the given set of param- eters. The biggest Lyapunov exponents determining whether a stable solution exists are shown in the Fig.2 for some anisotropy values. The mentioned phenomenon evidently follows from mode-mode interaction. It is absent at single- mode operation when at any value of amplitude anisotropy there is a stable solution with mode polarization in the lower loss plane. When anisotropy being further rised (up to about 100 kHz for the used set of parameters), the state with both modes polarized in the lower loss plane becomes stable on full duty cycle. Such situation is realized in a commercial laser LGN- 207B with Brewster’s window introducing considerable amplitude anisotropy into a cavity. B) Nonzero magnetic field In studying a laser in external magnetic field we were mainly interested in two questions: 1) do linear polarization planes remain orthogonal, and 2) how does magnetic field change their orientation. The set of equations (1)-(6) has been integrated for two values of the cavity anisotropy: 5·10-4 MHz and 1·10-1 MHz at different magnitudes of external magnetic field. It fol- lows from preceding consideration that in zero field at Qxy=5·10-4 MHz the state with modes polarized linearly in orthogonal planes is stable on almost full gain profile while an unstability range is very narrow, it is less then 6 MHz. At Qxy=5·10-4 MHz the state with both modes polarized in the lower loss plane is stable on full duty cycle. First of all, it is worth noting that a spatial mode always has nonzero ellipticity in magnetic field owing to nonequa- lity of left- and right-circularly polarized modes intensities. So, we will mean a direction of the bigger axis of a polariza- tion ellipse as a direction of a polarization vector. Calculated angles of the modes polarization planes and the interplane angle versus the middle intermode frequency ν12=(ν1+ν2)/2 (where ν1 and ν2 are modes operation fre- quencies) are shown in the Figs 3,4. The mode being called the first has the lower frequency while the mode being called the second has the higher one with respect to each other. It is worth noting that the interplane angle is close enough but not equal to that without field (i.e. to 0° or 90°) on the full duty cycle except the central tuning of one mode (when its frequency is close to the central transition frequency) at both values of anisotropy. At the central tuning the interplanes angle equals to that without field to within calculation er- rors. The polarization planes undergo turn in magnetic field. The fact is that the stronger anisotropy a cavity has the higher field magnitude is needed for that turn to be significant. In both considered cases the turn is more significant at the close to symmetrical tuning while it is small at the close to cen- tral tuning contrary to single-mode operation when turn of the polarization plane of a mode is greater at the central X X ω -D/20 ω +D/20ω0 Fig. 1. Direction of mode polarization plane versus its location at the gain line at weak anisotropy. Fig. 2. The determinative Lyapunov exponent versus the middle intermode frequency, (ν2+ν1)/2 at γab=225 MHz, γa=160 MHz, γb=290 MHz, η=1.33, Ku=1010 MHz, D=640 MHz and Qxy=0.5 kHz (a), Qxy=3.0 kHz (b), Qxy=6.0 kHz (c) without applied magnetic field. -300 -150 0 150 300 0 -3 -3 -3 -3 (π,0 ) (0 ,π) λc ν12, MHz λc ν12, MHz λc ν12, MHz -300 -150 0 150 300 0 -2 -2 -2 -2 (π,0 ) (0 ,π) -300 -150 0 150 300 0 -2 -2 -2 -2 -1 (π,0 ) (0 ,π) a) b) c) G.L. Kononchuk, S.M. Yegorov: Polarization unstabilities in a quasi-isotropic... 39SQO, 2(2), 1999 tuning. We can conclude that the mentioned phenomenon is a result of coupled action of the external magnetic field and mode-mode interaction. That is a reflection of a new type items in the set of equations which are absent in sin- gle-mode operation equations. Conclusions In the present paper a two-mode λ = 0.63 µm laser opera- tion with the presence both amplitude and phase anisotropy and axial magnetic field was considered on the basis of the general Lamb model. Numeric integration of the nonlinear equations set gave a number of time-independent solutions, and the Lyapunov stability analysis have been proceeded for them. It was found that without magnetic field at weak anisotropy the state with linearly polarized in orthogonal planes modes is stable at almost full duty cycle with the nearest to the centre mode is polarized in the lower loss plane while at strong enough anisotropy the state with both modes polarized in the lower loss plane becomes stable on full duty cycle. It turned out that even in zero magnetic field no time-independent state exists at the close to the sym- metrical tuning in the presence of weak amplitude anisotropy. The interplane angle and angles of the modes polarization planes was enumerated in nonzero field. It was found that changing orientation and nonorthogonality/ nonparallelity of the polarization planes takes place. They -300 -200 -100 0 0 30 60 90 120 0 1,2 ( + )/2, MHzn n1 2 ( + )/2, MHzn n1 2 0 2 1 ( )a b 0 1,2 -300 -150 0 150 300 83 84 85 86 87 88 89 90 -300 -150 0 150 300 0 1 2 3 4 5 6 ( + )/2, MHzn n1 2 ( + )/2, MHzn n1 2 0 2 1 | |a b Fig. 4. Calculated angles of the modes polarization planes φ1, φ2 (a) and the module of the interplane angle |φ2−φ1| (b) versus the middle intermode frequency, (ν2+ν1)/2. H=20G and the circled line corresponds to mode 1 and the crossed – to mode 2 in (a). The squared line corresponds to H=8G, the circled line – to H=12G, the plane line – to H=16G, the diamonded line – to H=20G. Qxy =100 kHz, η=1.38, and the other parameters are the same as in Fig.2. Fig. 3. Calculated angles of the modes polarization planes φ1, φ2 (a) and the interplane angle φ2−φ1 (b) versus the middle intermode frequency, (ν2+ν1)/2. The squared line corresponds to magnetic field H=0.2G, the circled line – to H=0.3G, the crossed line – to H=0.45G. Qxy=0.5 kHz, the other param- eters are the same as in Fig.2. G.L. Kononchuk, S.M. Yegorov: Polarization unstabilities in a quasi-isotropic... 40 SQO, 2(2), 1999 are more significant at the close to symmetrical tuning con- trary to single-mode operation when the turn of the polari- zation plane of a mode is greater at the central tuning. It was concluded that the reported phenomena belongs to the mode-mode interaction depending on the mode-mode phase correlations between them. Appendix α, β, θ, θ’ , σ, ρ, τ, τ’ can be evaluated on the basis of the paper [1]. In the present paper these values have been cal- culated in assumption of lifetimes equality of both atomic levels magnetic sublevels involved and equality of the Lande factors of both atomic levels involved, one isotop, no nu- clear spin and active medium filling all the cavity. σ is the real part while α is the imaginary one of the complex coef- ficient ΑΑΑΑΑ, ),Re( ),Im( nnnn ±±±± Α=Α= σα (A.1) where )1])([()( )8( 2 ''1' 3 0n −−±+× ×=Α ±± ± ℘∑ ν νε gHìiãçZ,ä Ku Bab,ba a,b ba' h ,(A.2) n = 1,2, ν is the operating frequency of the given mode, νn± is the detuning of the mode from the atomic line centre, g is the factor Lande, H is the magnitude of external magnetic field, ℘ab is an electric dipole matrix element, Ku is the Doppler parameter, γab is the Lorentz width, η is the mode gain, Z[x] is the plasma integral. Coefficients ρ, τ, τ’ are real parts while β, θ, θ’ are im- aginary ones of complex Θ’s. ),'Im(' ),Im( ),Im( ),Im( nnn,nn,n nnn,nn ±±′±′± ±±±±± Θ=Θ= Θ=Θ= θθ θβ mm (A.3a) ),'Re(' ),Re( ),Re( ),Re( nnn,nn,n nnn,nn ±±′±′± ±±±±± Θ=Θ= Θ=Θ= ττ τρ mm (A.3b) For different Θ’s we have: )()()8( 4 1 1 4 ''1'' 3 0n,n tk t= t,ba a,b ,ba íTKu ∑℘∑ ±±± =Θ δνεη h , (A.4) where n=1,2, the Tt1(νtk) are in the Doppler limit as fol- lows: 0,0,)]([2 32 1 3121 ≈≈+ − tttttt T Tíííði=T , (A.5) the arguments νtk can be found in the table A.1. }{ 4 1= 1 )2 4 1= 1 )1 4 ','1 3 0n,n )(+)( )()8( ∑∑ ∑ ℘ × ×=Θ ±±′± t tkt t tkt a,b baa',b' íTT Ku ν δνεη h (A.6) where n, n’= 1,2; n ≠ n’ (if n = n’ the former equation should be used). Let’s note two sums of the Ttk(νtk) functions in the k=1 k=2 k=3 t=1 )( ±± nab gH-íì+iã B aã )( ±nab gH-íì+iã Bm t=2 )( ±± nab gH-íì+iã B aã )( ±nab gH-íì+iã Bm t=3 )( ±± nab gH-íì+iã B bã )( ±+ nab ígHì+iã Bm t=4 )( ±± nab gH-íì+iã B bã )( ±nab gH-íì+iã Bm Table A.1. This table defines the arguments of Tt1 appearing in the third-order integrals ΘΘΘΘΘn±,n± (A.4) Σ1) k=1 k=2 k=3 t=1 )( ±± nab gH-íì+iã B aã )( ±nab gH-íì+iã Bm t=2 )( ±± nab gH-íì+iã B aã )( ±+ nab ígHì+iã Bm t=3 )( ±± nab gH-íì+iã B )( ±±′ nnb -íí+iã )( ±+ nab ígHì+iã Bm t=4 )( ±± nab gH-íì+iã B )( ±±′ nnb -íí+iã )( ±nab gH-íì+iã Bm Σ2) k=1 k=2 k=3 t=1 )( ±± nab gH-íì+iã B )( ±±′ nna -íí+iã )( ±± nab gH-íì+iã B t=2 )( ±± nab gH-íì+iã B )( ±±′ nna -íí+iã )( ±+ nab ígHì+iã Bm t=3 )( ±± nab gH-íì+iã B bã )( ±+ nab ígHì+iã Bm t=4 )( ±± nab gH-íì+iã B bã )( ±nab gH-íì+iã Bm Table A.2. This table defines the arguments of Tt1 appearing in the third-order integrals Qn±,n�± (A.6) G.L. Kononchuk, S.M. Yegorov: Polarization unstabilities in a quasi-isotropic... 41SQO, 2(2), 1999 Σ1) k=1 k=2 k=3 t=1 )( ±± nab gH-íì+iã B aã )( ±nab gH-íì+iã Bm t=2 )( ±± nab gH-íì+iã B aã )( ±+± nab ígHì+iã B t=3 )( ±± nab gH-íì+iã B )2( ±′± nnb -ígH+íì+iã B m )( ±+± nab ígHì+iã B t=4 )( ±± nab gH-íì+iã B )2( ±′± nnb -ígH+íì+iã B m )( ±± nab gH-íì+iã B Σ2) k=1 k=2 k=3 t=1 )( ±± nab gH-íì+iã B )2( ±′± nna -ígH+íì+iã B m )( ±± nab gH-íì+iã B t=2 )( ±± nab gH-íì+iã B )2( ±′± nna -ígH+íì+iã B m )( ±+± nab ígHì+iã B t=3 )( ±± nab gH-íì+iã B bã )( ±+± nab ígHì+iã B t=4 )( ±± nab gH-íì+iã B bã )( ±+± nab ígHì+iã B Table A.3. This table defines the arguments of Tt1 appearing in the third-order integrals Qn±,n� m (A.7) Σ1) k=1 k=2 k=3 t=1 )( ±′′± nnnab -í+ígH-íì+iã B mm )( ±′ nna -íí+iã m )( ±nab gH-íì+iã Bm t=2 )( ±′′± nnnab -í+ígH-íì+iã B mm )( ±′ nna -íí+iã m )( ±+± nab ígHì+iã B t=3 )( ±′′± nnnab -í+ígH-íì+iã B mm )2( ±′± nnb -ígH+íì+iã B m )( ±+± nab ígHì+iã B t=4 )( ±′′± nnnab -í+ígH-íì+iã B mm )2( ±′± nnb -ígH+íì+iã B m )( ±± nab gH-íì+iã B Σ2) k=1 k=2 k=3 t=1 )( ±′′± nnnab -í+ígH-íì+iã B mm )2( ±′± nna -ígH+íì+iã B m )( ±± nab gH-íì+iã B t=2 )( ±′′± nnnab -í+ígH-íì+iã B mm )2( ±′± nna -ígH+íì+iã B m )( ±+± nab ígHì+iã B t=3 )( ±′′± nnnab -í+ígH-íì+iã B mm )( ±′ nnb -íí+iã m )( ±+± nab ígHì+iã B t=4 )( ±′′± nnnab -í+ígH-íì+iã B mm )( ±′ nnb -íí+iã m )( ±nab gH-íì+iã Bm Table A.4. This table defines the arguments of Tt1 appearing in the third-order integrals Q�n± (A.8) References 1. M. Sargent III, W. E. Lamb, R. L. Fork, Theory of a Zeeman Laser I, II // Phys. Rev. 164(2), pp.436-465 (1964) 2. D. Lenstra, On the theory of polarization effects in gas laser // Phys. Reports 59(3), pp.301-373 (1980) 3. L. P. Svirina, Polarization unstability in a two-frequency gas laser with a weakly anisotropical cavity // Optika i spektroscopiya 77(1), pp.124-133 (1994) (in Russian). brackets. This is reflected in the following table A.2 which consists of two parts for the first and for the second sum. } { 4 1 )22 2 2 '' 4 1 )1 2 2 2 ''1'' 3 0n,n )()()(+)( )()()8( 11 ∑℘℘∑ ℘℘∑ × ×=Θ ±±′± t= tka',b',ba t= tk a',b',ba a,b ,ba tt TT äKu νν νεη m m h (A.7) where n, n’ = 1,2, the arguments νtk are in the table A.3. ×=Θ′ ±±± ℘℘∑ 2 2'' 2 ''1'' 3 0n )()()8( { ,ba,ba a,b ,baäKuhνεη } 4 1 )(1 )22 2'' 2 '' 4 1 )(1 )1 )()( ∑℘℘∑× t= tkt,ba,ba t= tkt íT+íT m (A.8) where n = 1,2, the arguments νtk are in the table A.4.

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