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Analogy between higher instabilities in fluids and lasers
Volume 53A, number 1
PHYSICS LETTERS
19 May 1975
A N A L O G Y B E T W E E N H I G H E R I N S T A B I L I T I E S IN F L U I D S A N D L A S E R S H. HAKEN Institut flit theoretische Physik der Universitd'tStuttgart, Germany Reoeived 10 April 1975 The Lorenz model of instabilities in fluids is shown to be identical with that of the single mode laser and applicable to undamped laser spikes. Further instabilities connected with small-band excitations are also discussed. It is known that the continuous mode laser close to the laser threshold and a fluid close to the convection instability can be described by Ginzburg-Landau equations [1,2]. The reasons for these results which establish a close analogy between lasers and fluids follow from rather general principles and are discussed elsewhere %. Recently, we have found a very detailed analogy between fluids and lasers which goes far beyond what one could expect on general grounds %. Beyond the interest by its own, this analogy has immediate important applications to undamped spiking in lasers and masers. Our investigation was triggered by recent papers on instabilities of fluids by McLaughlin and Martin [4] and we closely follow their notation. We start from the usual equations of fluid dynamics in the Boussinesq approximation [e.g. 5, 6] which contain as variables the velocity field u = (Ul, u2, u3) and the deviation 0 from a constant temperature graclient. We use dimensionless units and expand u into Fourier series:
u/(x, y,
z) =
(1)
= i/,m, n~=_® u/(l, m, n) exp {i(kllx+k2my+ncrz )}
0(0, 0, 2) = 0002. (This choice of components corresponds to a motion of convection roils in the x-z plane). Inserting (1) and (2) into the above mentioned equations one obtains (after a slight "renormalization" of variables (see ref. [4]): t~101 = o0101 -- aUl01
(3) ,
2
+ [{ioa(a/ax) + oa V 2} Ulol], /9101 = +
-
Ulol°o02
+ rUlol
-
°101
(4)
[{ia(~)/Ox) +a'V2}Olox ] ,
•
'
2
0002 = U1010101 - b0002 + [a V20002].
(5)
o = v/r' is the Prandtl number (where v is the kinematic viscosity, r ' the thermometric conductivity); r = R / R e (where R is the Rayleigh number, R e the critical Rayleigh number), b = 41r2/0r 2 + k~). The terms in brackets have been added by us. They occur if we allow for slowly varying amplitudes u/(l, m, n), 0(l, m, n), in the sense of Newel1 and Whitehead [8]. a = 2 k l ( k l2 +Ir2) -1 , a' =(k21 + ~.2)-1, and
and 0 correspondingly.
(2)
x, y are the horizontal coordinates and k 1, k 2 the corresponding fundamental wave vectors, z is the vertical coordinate. To obtain the Lorenz model [7], we retain only Ul(l, 0, I) --- Ul01,0(I, 0, I) = 0101, * A brief account was given at the Spring Meeting of the German Phys. Society, Miinster, March 1975.
V2 _- a2/ax 2 + o2/ay2. We first neglect the terms in brackets and put Ul01 = ~, 0101 = ¢/, 0002 = r - ~', which yields
We now describe the laser equations assuming the laser field propagating in x direction and polarized. We decompose the field strength E as 77
Volume 53A, number 1
PHYSICS LETTERS
E(x, t) = E~(x, t) exp {iW o ( t - x / c ) } + c.c. and the polarization P correspondingly. 600 is the atomic transition frequency and c the light velocity in the medium. Using the rotating wave approximation and the slowly varying amplitude approximation, the laser equations read [9] g
= KP
-
KE
-
[c~E/~x]
(7)
$ e = 3'ED - 3'J~
(8)
/~ = 3'jl(h+ 1) - 3,110 -3'U ~ p "
(9)
Following~ RJ~sken and Nummedal [10], we have normalized E, P, and the inversion D by dividing /~ ~, D by their c.w. values, r is the cavity loss, 3' the linewidth, 3'11the inverse longitudinal relaxation time, X = (D o - Dc)/D c, where Do, D c are the unsaturated and critical inversion, respectively. If 3E/Ox = 0, eqs. ( 7 ) - ( 9 ) reduce to those of the single mode laser, if collective atomic coordinates are used and phases put = 0 [11,12]. In this case, eqs. ( 7 ) - ( 9 ) are identical with that of the Lorenz model in the form (6), which can be realized by the following identifications:
t "-~t'tT/K, /~--~ Ot~, where a = { b ( r - 1 ) } -1/2, r > l , P-~ arl, D-~ ~, 3"11= rb/e, 3" = r/o, h = r - 1. Eqs. (6)describe at least two instabilities which have been found independently in fluid dynamics and in lasers. For ?~< 0 (r < 1) there is no laser action (the fluid is at rest), for ?~>0 (r~> 1) laser action (convective motion starts) with stable, time-independent solutions ~, r?, ~'. Besides this well known "classical" instability, a new one occurs provided K > 3' + 3'11, ( o > b + 1) and h > (7+7tl +K)(3"+K)/7(K--3"--3"11), {r > a ( o + b + 3 ) / ( o - 1 - b ) ) *. (For lasers see [12], for fluids [4, 7]). While it has been noted [12] that this instability gives rise to undamped laser spikes, its detailed features remained unexplained in this field. In fluid dynamics, its nature could be clarified * Note that in both cases the parameters h and r can be experimentally varied (h by the pump, r by the temperature gradient).
78
19 May 1975
to a large extent by a machine calculation by Lorenz [7] and a detailed analysis using the concept of "inverted bifurcation" by McLaughlin and Martin [4]. The most important result is that spiking occurs randomly though the equations are completely deterministic. (For more details see [4, 7]). While the above instability condition cannot be met in realistic fluids, it can be fulfilled in lasers and masers in high-loss cavities. In conclusion, we mention the impact of the terms in brackets in (7) and (3). They give rise to a different class of instabilities [1, 10] which have been treated by a machine calculation [10] and by a generalized bifurcation theory (including noise) [3]. Though the first order derivation in (3) differs from that of (7) by the imaginary unit, both terms cause an instability at corresponding values of ?, and r. A detailed discussion which uses, among others, a mode expansion of E and/~ of the form (1) will be published elsewhere, relating it to [4] and work by Busse [13].
References [1] R. Graham and H. Haken, Z. Physik 210 (1968) 276; 237 (1970) 31; H. Haken, in Advances in solid state physics, ed. O. Madelung (Vieweg-Pergamon, 1970). [2] R. Graham, Phys. Rev. Lett. 31 (1973) 1479; Phys. Rev. 10 (1974) 1762; H. Haken, Phys. Lett. A46 (1973) 193; Rev. Mod. Phys. 47 (1975) 67. [3] H. Haken, Z. Physik, in press. [4] J.B. McLaughlin and P.C. Martin, Phys. Rev. Lett. 33 (1974) 1189, and preprint 1975. [5] A.S. Monin and A.M. Yagiom, Statistical fluid mechanics, Vol. I (M.I.T. Press, 1971). [6] H. Haken, ref. [2]. [7] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [8] A. Newell and J. Whitehead, J. Fluid Mech. 38 (1969) 279. [9] H. Haken, Laser theory, Encyclopedia of Physics, XXV/2c (Springer, New York, 1970). [10] H. Risken and K. Nummedal, Phys. Lett. 26A (1968) 275; J. Appl. Phys. 39 (1968) 4662. [11] H. Haken and H. Sauermann, Z. Physik 173 (1963) 261; 176 (1963) 47; H. Haken, Z. Physik 181 (1964) 96. [12] H. Haken, Z. Physik 190 (1966) 327. [13] F.H. Busse, J. Fluid Mech. 52 (1972) 1, 97.
PHYSICS LETTERS
19 May 1975
A N A L O G Y B E T W E E N H I G H E R I N S T A B I L I T I E S IN F L U I D S A N D L A S E R S H. HAKEN Institut flit theoretische Physik der Universitd'tStuttgart, Germany Reoeived 10 April 1975 The Lorenz model of instabilities in fluids is shown to be identical with that of the single mode laser and applicable to undamped laser spikes. Further instabilities connected with small-band excitations are also discussed. It is known that the continuous mode laser close to the laser threshold and a fluid close to the convection instability can be described by Ginzburg-Landau equations [1,2]. The reasons for these results which establish a close analogy between lasers and fluids follow from rather general principles and are discussed elsewhere %. Recently, we have found a very detailed analogy between fluids and lasers which goes far beyond what one could expect on general grounds %. Beyond the interest by its own, this analogy has immediate important applications to undamped spiking in lasers and masers. Our investigation was triggered by recent papers on instabilities of fluids by McLaughlin and Martin [4] and we closely follow their notation. We start from the usual equations of fluid dynamics in the Boussinesq approximation [e.g. 5, 6] which contain as variables the velocity field u = (Ul, u2, u3) and the deviation 0 from a constant temperature graclient. We use dimensionless units and expand u into Fourier series:
u/(x, y,
z) =
(1)
= i/,m, n~=_® u/(l, m, n) exp {i(kllx+k2my+ncrz )}
0(0, 0, 2) = 0002. (This choice of components corresponds to a motion of convection roils in the x-z plane). Inserting (1) and (2) into the above mentioned equations one obtains (after a slight "renormalization" of variables (see ref. [4]): t~101 = o0101 -- aUl01
(3) ,
2
+ [{ioa(a/ax) + oa V 2} Ulol], /9101 = +
-
Ulol°o02
+ rUlol
-
°101
(4)
[{ia(~)/Ox) +a'V2}Olox ] ,
•
'
2
0002 = U1010101 - b0002 + [a V20002].
(5)
o = v/r' is the Prandtl number (where v is the kinematic viscosity, r ' the thermometric conductivity); r = R / R e (where R is the Rayleigh number, R e the critical Rayleigh number), b = 41r2/0r 2 + k~). The terms in brackets have been added by us. They occur if we allow for slowly varying amplitudes u/(l, m, n), 0(l, m, n), in the sense of Newel1 and Whitehead [8]. a = 2 k l ( k l2 +Ir2) -1 , a' =(k21 + ~.2)-1, and
and 0 correspondingly.
(2)
x, y are the horizontal coordinates and k 1, k 2 the corresponding fundamental wave vectors, z is the vertical coordinate. To obtain the Lorenz model [7], we retain only Ul(l, 0, I) --- Ul01,0(I, 0, I) = 0101, * A brief account was given at the Spring Meeting of the German Phys. Society, Miinster, March 1975.
V2 _- a2/ax 2 + o2/ay2. We first neglect the terms in brackets and put Ul01 = ~, 0101 = ¢/, 0002 = r - ~', which yields
We now describe the laser equations assuming the laser field propagating in x direction and polarized. We decompose the field strength E as 77
Volume 53A, number 1
PHYSICS LETTERS
E(x, t) = E~(x, t) exp {iW o ( t - x / c ) } + c.c. and the polarization P correspondingly. 600 is the atomic transition frequency and c the light velocity in the medium. Using the rotating wave approximation and the slowly varying amplitude approximation, the laser equations read [9] g
= KP
-
KE
-
[c~E/~x]
(7)
$ e = 3'ED - 3'J~
(8)
/~ = 3'jl(h+ 1) - 3,110 -3'U ~ p "
(9)
Following~ RJ~sken and Nummedal [10], we have normalized E, P, and the inversion D by dividing /~ ~, D by their c.w. values, r is the cavity loss, 3' the linewidth, 3'11the inverse longitudinal relaxation time, X = (D o - Dc)/D c, where Do, D c are the unsaturated and critical inversion, respectively. If 3E/Ox = 0, eqs. ( 7 ) - ( 9 ) reduce to those of the single mode laser, if collective atomic coordinates are used and phases put = 0 [11,12]. In this case, eqs. ( 7 ) - ( 9 ) are identical with that of the Lorenz model in the form (6), which can be realized by the following identifications:
t "-~t'tT/K, /~--~ Ot~, where a = { b ( r - 1 ) } -1/2, r > l , P-~ arl, D-~ ~, 3"11= rb/e, 3" = r/o, h = r - 1. Eqs. (6)describe at least two instabilities which have been found independently in fluid dynamics and in lasers. For ?~< 0 (r < 1) there is no laser action (the fluid is at rest), for ?~>0 (r~> 1) laser action (convective motion starts) with stable, time-independent solutions ~, r?, ~'. Besides this well known "classical" instability, a new one occurs provided K > 3' + 3'11, ( o > b + 1) and h > (7+7tl +K)(3"+K)/7(K--3"--3"11), {r > a ( o + b + 3 ) / ( o - 1 - b ) ) *. (For lasers see [12], for fluids [4, 7]). While it has been noted [12] that this instability gives rise to undamped laser spikes, its detailed features remained unexplained in this field. In fluid dynamics, its nature could be clarified * Note that in both cases the parameters h and r can be experimentally varied (h by the pump, r by the temperature gradient).
78
19 May 1975
to a large extent by a machine calculation by Lorenz [7] and a detailed analysis using the concept of "inverted bifurcation" by McLaughlin and Martin [4]. The most important result is that spiking occurs randomly though the equations are completely deterministic. (For more details see [4, 7]). While the above instability condition cannot be met in realistic fluids, it can be fulfilled in lasers and masers in high-loss cavities. In conclusion, we mention the impact of the terms in brackets in (7) and (3). They give rise to a different class of instabilities [1, 10] which have been treated by a machine calculation [10] and by a generalized bifurcation theory (including noise) [3]. Though the first order derivation in (3) differs from that of (7) by the imaginary unit, both terms cause an instability at corresponding values of ?, and r. A detailed discussion which uses, among others, a mode expansion of E and/~ of the form (1) will be published elsewhere, relating it to [4] and work by Busse [13].
References [1] R. Graham and H. Haken, Z. Physik 210 (1968) 276; 237 (1970) 31; H. Haken, in Advances in solid state physics, ed. O. Madelung (Vieweg-Pergamon, 1970). [2] R. Graham, Phys. Rev. Lett. 31 (1973) 1479; Phys. Rev. 10 (1974) 1762; H. Haken, Phys. Lett. A46 (1973) 193; Rev. Mod. Phys. 47 (1975) 67. [3] H. Haken, Z. Physik, in press. [4] J.B. McLaughlin and P.C. Martin, Phys. Rev. Lett. 33 (1974) 1189, and preprint 1975. [5] A.S. Monin and A.M. Yagiom, Statistical fluid mechanics, Vol. I (M.I.T. Press, 1971). [6] H. Haken, ref. [2]. [7] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [8] A. Newell and J. Whitehead, J. Fluid Mech. 38 (1969) 279. [9] H. Haken, Laser theory, Encyclopedia of Physics, XXV/2c (Springer, New York, 1970). [10] H. Risken and K. Nummedal, Phys. Lett. 26A (1968) 275; J. Appl. Phys. 39 (1968) 4662. [11] H. Haken and H. Sauermann, Z. Physik 173 (1963) 261; 176 (1963) 47; H. Haken, Z. Physik 181 (1964) 96. [12] H. Haken, Z. Physik 190 (1966) 327. [13] F.H. Busse, J. Fluid Mech. 52 (1972) 1, 97.