Файл: Staliunas K., Sanchez-Morcillo V.J. (eds.) Transverse Patterns in Nonlinear Optical Resonators(ST.pdf
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2.3 Derivation of the Laser Order Parameter Equation |
47 |
us to consider cases where ω and 2 are moderate or large, as long as their di erence is small.
We make also the near-to-threshold assumption,
D0 = 1 + p , p = µε2 . |
(2.30) |
With these assumptions, the laser variables depend on slow temporal and spatial scales, which can be determined by using the results of the linear stability analysis. The eigenvalue (2.8), which determines the temporal evolution in the linear stage, has terms of first and second order in ε. We can then define two temporal scales,
T1 = εt , T2 = ε2t , |
(2.31) |
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which allows us to expand the temporal derivative as |
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∂ |
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(2.32) |
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∂T1 |
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∂t |
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∂T2 |
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Finally, we expand the fields around the trivial solution:
∞ |
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E = |
εnen , P = |
εnpn , D = D0 + εndn . |
(2.33) |
n=1 |
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All the definitions (2.29)–(2.33) are now introduced into the Maxwell– Bloch equations (2.1). Powers of ε are gathered, and the equations are solved recursively at each order.
At the first order,
e1 = p , d1 = 0 .
At the second order,
1 ∂e1 = −e2 + p2 + iΘe1 , κ ∂T1
1 ∂p1 = e2 − p2 , γ ∂T1
0 = d2 − 12 (e1p1 + p1e1) .
The compatibility of (2.35a) and (2.35b) requires that
∂e1 |
= i |
κγ |
Θe1 , |
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∂T1 |
κ + γ |
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while the polarization is related to the field through
κ
p2 = e2 − iκ + γ Θe1 .
(2.34)
(2.35a)
(2.35b)
(2.35c)
(2.36)
(2.37)
48 2 Order Parameter Equations for Lasers
Both of these relations will be useful later, at the next order. From (2.35c), we obtain
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d2 = − |e1|2 . |
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(2.38) |
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At the order O(ε3), the equations read |
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1 ∂e1 |
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1 ∂e2 |
= −e3 + p3 + iΘe2 , |
(2.39a) |
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κ ∂T2 |
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∂T1 |
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1 ∂p1 |
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1 ∂p2 |
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+ |
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= e3 − p3 + µe1 + d2e1 . |
(2.39b) |
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∂T2 |
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∂T1 |
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This system reduces, by applying a solvability condition that actually consists in eliminating the dependence on third-order contributions, to
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κ2 |
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= µe1 −e1 |e1|2 +iΘe2 − |
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Θ2e1 , (2.40) |
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γ |
∂T2 |
∂T1 |
(κ + γ )2 |
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where, in obtaining the last term, we have used |
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1 ∂p2 |
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1 ∂e2 |
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κ |
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∂e1 |
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κ2 |
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= |
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− |
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iΘ |
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Θ2e1 . (2.41) |
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∂T |
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∂T |
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κ + γ |
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γ |
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∂T |
1 |
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(κ + γ ) |
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Equation (2.40) depends now only on the field amplitude at di erent orders. Let us now define an order parameter A = εe1 + ε2e2. The evolution of the order parameter with respect to the original time t is given by
∂t |
= ε |
∂T1 + ε |
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∂T1 |
+ ∂T2 |
= ε |
κ + γ iΘe1 |
+ ε |
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∂T1 |
+ ∂T2 . |
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∂A |
2 |
∂e1 |
3 |
∂e2 |
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∂e1 |
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2 κγ |
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(2.42) |
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Finally, (2.42), expressed in terms of the original variables, gives the evolution equation of the order parameter,
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∂A |
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+ |
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= (D0 − 1) A − A |A| |
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κ |
γ |
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∂t |
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+i a 2 − ω A − |
κ2 |
a 2 − ω |
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A , (2.43) |
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(κ + γ )2 |
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which coincides with (2.26), obtained by using the adiabatic-elimination procedure.
References
1.M.C. Cross and P.C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). 33
References 49
2.R. Graham and H. Haken, Laserlight – first example of a second order phase transition far from thermal equilibrium, Z. Phys. 237, 31 (1970). 34, 41
3.K. Staliunas, Laser Ginzburg–Landau equation and laser hydrodynamics, Phys. Rev. A 48, 1573 (1993). 41
4.J. Lega, J.V. Moloney and A.C. Newell, Swift–Hohenberg equation for lasers, Phys. Rev. Lett. 73, 2978 (1994). 41, 46
5.J.B. Swift and P.C. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A 15, 319 (1977). 45
6.L.A. Lugiato and R. Lefever, Spatial dissipative structures in passive optical systems, Phys. Rev. Lett. 58, 2209 (1987). 46
3 Order Parameter Equations for Other Nonlinear Resonators
3.1 Optical Parametric Oscillators
An optical parametric oscillator basically consists of a nonlinear χ(2) medium
¯
inside a resonator driven by a coherent field of amplitude E and frequency ωL, which propagates along the optical axis of the resonator, parallel to the z axis. The crystal converts the intracavity pump field of frequency ωL and amplitude A0 into two fields of frequency f1ωL and f2ωL, and of amplitude A1 and A2, the signal and idler waves, respectively. Energy conservation requires that f1 + f2 = 1. Three longitudinal modes of the cavity with frequencies ωcm (m = 0, 1, 2) are assumed to be close to the frequencies fmωL (where f0 = 1). Under these conditions, and making some of the usual assumptions of nonlinear optics (the mean-field limit, the paraxial and single-longitudinal- mode approximations), the evolution equations for the pump, signal and idler fields can be written as [1]
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∂A0 |
= γ0 |
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(1 + iω0) A0 |
¯ |
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2 |
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(3.1a) |
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− |
+ E |
− |
A1A2 + ia0 |
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∂t |
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= γ1 |
− |
(1 + iω1) A1 |
+ A0A2 + ia1 |
2A1 |
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(3.1b) |
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∂A2 |
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∂t |
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= γ2 |
− (1 + iω2) A2 |
+ A0A1 + ia2 cA2 |
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(3.1c) |
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∂t |
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ω |
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where γm are |
the cavity decay rates, ω |
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= (ω |
m |
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f |
m |
L |
m |
are the detun- |
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2 |
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ings and am = c |
/2γmfmωL are the di raction coe cients. |
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The signal and idler fields can have arbitrary frequencies, since f1 and |
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f2 |
are free (within the restriction f1 + f2 = 1). In the particular case f1 = |
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f2 |
= 1/2, both fields have the same frequency ωL/2, leading to degenerate |
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oscillation (the DOPO). In this case, the model takes the form |
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∂A0 |
= γ0 |
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(1 + iω0) A0 |
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(3.2a) |
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− |
+ E |
− |
A1 + ia0 |
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A0 |
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∂A1 |
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∂t |
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= γ1 |
− (1 + iω1) A1 |
+ A0A1 + ia1 2A1 |
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(3.2b) |
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∂t |
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Note that the degenerate model follows from the condition X1 = X2, where X is any of the variables. One might think that, in principle, the results from the nondegenerate model would include also those corresponding to the
K. Stali¯unas and V. J. S´anchez-Morcillo (Eds.):
Transverse Patterns in Nonlinear Optical Resonators, STMP 183, 51–64 (2003)c Springer-Verlag Berlin Heidelberg 2003