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~ ~ ~ ~ ~ ~~ ~ , 3 + ( + + 1 i ) 2 + [ ( A + ) i (1 + )] + 2 ( A 1) i A = 0 in .NET Integration GTIN - 13 in .NET ~ ~ ~ ~ ~ ~~ ~ , 3 + ( + + 1 i ) 2 + [ ( A + ) i (1 + )] + 2 ( A 1) i A = 0

~ ~ ~ ~ ~ ~~ ~ , 3 + ( + + 1 i ) 2 + [ ( A + ) i (1 + )] + 2 ( A 1) i A = 0 using .net todraw ean-13 in asp.net web,windows application Barcode FAQs (4.146) where EAN13 for .NET is a real quantity and = i + , which corresponds to the Hopf type of the assumed bifurcation at the second laser threshold.

We then use. Fundamentals of Laser Dynamics the familiar me European Article Number 13 for .NET thod: only the term linear with respect to are related in Eq. (4.

146). Equation (4.146) is split into two real ones, which, after elimination of , reduce to.

~ = . ~ ~ ~ 4 [3 EAN-13 Supplement 5 for .NET ( A 1) ] 2 + 2 2 A( A 1) ~ ~ ~ ~ ~~ ~ ~ ~. 4 + [ (2 +1) + ( +1)2 2 A] 2 + 2 ( + 1)( A 1) + 2 A( A + ).

(4.147) The bou ndary between the domains of stable and unstable solutions of Eqs. (4.

144) can easily be found assuming = 0. The roots of the resultant quadratic equation for the case are given by. ~ 2 2 / = 3( UPC-13 for .NET A 1) 9( A 1) 2 8 A( A 1) ..

(4.148). They are real i ean13+5 for .NET f A Acr = 9 . Thus, the critical value of the pumping parameter for a multimode travelling-wave laser coincides with that obtained for ~ ~ a single-mode laser under the most favourable conditions << 1, = 3 .

The dependence expressed by Eq. (4.148) is given in Fig.

4.12. Its asymptotes for A >> 1 are represented by the lines.

~ ~ 2 / = 2 A, 2 / = A . max min (4.149). The potential i nstability domain is shown hatched in Fig. 4.12.

It should be borne in mind, however, that the instability could develop only if one frequency of the intermode beats enters this domain (the cycling condition). This additional (to A > Acr) requirement is a compensation for no limit being imposed on the cavity Q-factor. Roughly speaking, it is necessary that the normalized intermode frequency spacing = 2 c / L does ~ not exceed the frequency max = ( 2 A )1 / 2 .

Thus, a limit is placed on the cavity perimeter:. L > Lcr = 2 c( . A) 1/ 2 .. 4 -1. (4.150). Numerical estim ean13 for .NET ation for a Nd:YAG laser ( . = 10 s , = 10 12 s -1 ) for A = 10 yields max = max = 4 108 s -1 and Lcr 6 m. In principle, the fibre optical delay permits design of laser cavities with a perimeter of a few hundreds of meters [86], but these are rare in practice. So, this type of instability is not an urgent problem for solid-state laser.

However, it is easy to reach these values in is retained. If we take into account the existing spatial nonuniformity, caused, for example, by the loss localization on the mirrors, then the second threshold is still higher [396, 397]. Specific features of a multimode laser are essential to pulsations above the second threshold.

Since the spectrum of allowed pulsation frequencies is rigidly subject to the discrete spectrum of the cavity modes, the transient process from amplitude modulation to beats, shown in Fig. 3.6, is highly problematic.

Nevertheless, the numerical investigation of Eqs. (4.144) shows that higher-order bifurcations can occur.

The pulse train. Multimode Lasers with Frequency-Nondegenerate Modes Fig. 4.12.

Phas e diagram of a travelling wave multimode laser in the control parameter plane (A, ) plotted according to Eq. (4.148).

The hatching denotes the instability domain.. envelope in the visual .net EAN 13 laser instability domain can be both regular and chaotic [391, 395]. There is no strong experimental confirmation of the Risken NummedalGraham Haken theory until now.

Information about observation of undamped pulsations in an erbium fibre laser is reported in Ref. [398]. The period of observed pulsations is equal to the round trip time of the cavity.

According to all parameters this regime corresponds to the theory predictions. Only the instability threshold is evidently lower than theoretically predicted. This fact forced us to think about the modification of the model [399] or about the presence of any casual nonlinearity in the experiment, for example, a weak saturable absorption.

But one must bear in mind that the gain line can be slightly inhomogeneously broadened..
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