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L L M 0 t 1 = RM 1 t . c c in .NET Incoporate EAN-13 in .NET L L M 0 t 1 = RM 1 t . c c

L L M 0 t 1 = RM 1 t . c c generate, create none none in none projects iOS (7.109). Two equations, (7.108) and (7.109), can be combined into one equation.

Using the dimensionless variables. = . t , m = B 1 M ,. we write this equation as A = BN ( 0 ) L1 /(c ln R 1 ), 0 = L / c , (7.110). Lasers with Nonlinear Parameters Fig. 7.14.

Sche me of a ring laser with nonlinear absorbing cell and a selective element which represents the dispersive properties of the laser medium: 1 output mirror, 2,3 total reflecting mirrors, 4 laser rod, 5 nonlinear cell, 6 linear bandpass. m( ) 1 + none none m( ) Rm( 0 ) = A ln . (7.111) 1 + ln R Rm( 0 ).

The nonlinear differential-difference Eq. (7.111) has the unique steadystate solution ln R 1 ( A 1) . 1 R (7.112). If the losses a none none re small, such that 1 R << 1 , then Eq. (7.112) is slightly simplified: m = A 1.

Note that Eq. (7.111) does not have a trivial solution, since it was assumed that M 0 when this equation was derived.

In the vicinity of the steady state Eq. (7.111) can be linearized with respect to the variable m = m m .

The solutions of the linearized equation have the form m0 exp( ) , where l is the root of the characteristic equation. First we will find the roots, which satisfy the inequality 0 << 1 and admit an expansion exp( 0 ) = 1 + 0 . Eq. (7.113) transforms to + 1 + Rm = exp( 0 ) . +1+ m (7.113). 2 + ( m + 1) + (1 R ) 0 1m = 0 . Note that (7.114). c 1 R 2 = =G (1 R ) 0 L and, therefore, for 1 R<<1 Eq. (7.114) coincides with Eq.

(3.19)..

(7.115). Fundamentals of Laser Dynamics We now consider the roots of Eq. (7.113), which satisfy the condition >> 1 . Wh en one of these roots is substituted into Eq. (7.

113), the lefthand side of the equality is close to unity. This means that. ~ where i + << 0 1 none for none . Multiplying Eq. (7.

113) by the complex conjugate equation we find:. ~ = i + = 2 iq 0 1 + i + ,. 2 + ( + 1 + none for none Rm ) 2 = exp( 2 0 ) 1 + 2 0 , 2 + ( + 1 + m ) 2. hence 2 0 . m (1 R )[2 + m (1 R )] . (1 + m ) 2 + 2 (7.116). For q = 1, 2,... the first term in the denominator can be neglected so that 1 m (1 R )[2 none for none + m (1 R )] , 2 0 2. (7.117). and in the case none for none R 1, we find, in view of (7.115). = G 2 A( A 1) .. (7.118). This equation y ields a monotonic decrease of the perturbation damping rate as the perturbation frequency grows. This result is true as long as the rate equation approach, which ignores the laser medium dispersion, is valid. Taking the dispersion into account is critical exactly in the domain of highfrequency perturbation.

To see this we assume, following [574], that the laser cavity contains, besides the nondispersive laser medium, a hypothetical frequency filter with the passband to transform the laser intensity by the law. 1 Lfilt 1 + M out (t ) = M in t c . (7.119). Using this equation instead of Eq. (7.109) we obtain the set of equations m1 ( + 1 ) 1 + m1 ( + 1 ) m0 ( ) = Aln , (7.120a) 1 + ln m0 ( ) R ~ 1 + m0 ( + 0 ) = Rm1 ( + 1 ) . (7.120b). instead of (7.111). Here, i = ( Li L0 ) is none for none the dimensionless coordinate of each cross-section shown in Fig. 7.14 and mi is the field intensity in.

Lasers with Nonlinear Parameters the corresponding cross-section. Linearization of Eqs. (7.120) yields a characteristic equation + 1 + m0 = ex p( 0 ) . ~ ( + 1 + m1 )( + 1). (7.121). Multiplying (7. 121) by the complex-conjugate equation and neglecting the small terms we find the perturbation decay rate as a function of the perturbation frequency:. 1 = 2 0. 2 + (1 + m0 ) 2 2 2 ~ 2 + 1) 1 . [ + (1 + m1 ) ]( . (7.122). This dependence has an extremum for the frequency ~ max = 1 none for none / 2 [( 2 + m1 + m0 )( m1 m0 )1 / 4 .. (7.123).
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